Estimates for the Green's function and parameters of exponential dichotomy of a hyperbolic operator semigroup and linear relations

Let $\mathscr{H}$ be a complex Hilbert space, and $\operatorname{End} \mathscr{H}$ the Banach algebra of linear bounded operators acting in $\mathscr{H}$. We consider a strongly continuous operator semigroup (semigroup of class $C_{0}$)

$$\begin{equation*} T\colon \mathbb{R}_{+}=[0,\infty) \to \operatorname{End}\mathscr{H}, \end{equation*}$$

and let $A\colon D(A) \subset \mathscr{H} \to \mathscr{H}$ be its infinitesimal operator (see [1]).

We consider the linear differential equation

$$\begin{equation} \dot{x}(t)-A x(t)=f(t),\qquad t \in \mathbb{R}, \end{equation} \tag{ 1.1 }$$

where the function $f\colon \mathbb{R} \to \mathscr{H}$ belongs to one of the following Banach function spaces. Let $C_{b}=C_{b}(\mathbb{R},\mathscr{H})$ denote the Banach space of bounded continuous functions defined on $\mathbb{R}$ with values in $\mathscr{H}$ with the norm $\|x\|_{\infty}=\sup_{t \in \mathbb{R}}\|x(t)\|$. The symbol $L^{p}=L^{p}(\mathbb{R},\mathscr{H})$, $p \in [1, \infty]$, is used for denoting the Banach space of (classes of) Bochner measurable functions $x\colon \mathbb{R} \to \mathscr{H}$ for which the following quantity is finite:

$$\begin{equation*} \|x\|_{p}=\biggl(\displaystyle\int_{\mathbb{R}}\|x(t)\|^{p}\,dt\biggr)^{1/p}, \quad p \ne \infty, \qquad \|x\|_{\infty}=\operatorname{ess\,sup}_{t\in\mathbb{R}}\|x(t)\|, \quad p=\infty. \end{equation*}$$

If $p=2$, then $L^{2}(\mathbb{R},\mathscr{H})$ is a Hilbert space with the inner product $\langle x,y\rangle= \displaystyle\int_{\mathbb{R}}(x(t),y(t))\,dt$, $x,y \in L^{2}$. We use the symbol $\mathscr{F}=\mathscr{F}(\mathbb{R},\mathscr{H})$ to denote one of the spaces $C_{b}(\mathbb{R},\mathscr{H}),L^{p}(\mathbb{R},\mathscr{H})$, $p \in [1,\infty]$.

A weak bounded solution of equation (1.1), where $f \in \mathscr{F}(\mathbb{R},\mathscr{H})$, is defined to be a function $x \in C_{b}\cap \mathscr{F}$ such that the equations

$$\begin{equation} x(t)=T(t-s)x(s)+\int_s^t T(t-\tau)f(\tau)\,d\tau,\qquad s \leqslant t, \end{equation} \tag{ 1.2 }$$

hold for all $s \leqslant t$ in $\mathbb{R}$.

The following problem is important: find necessary and sufficient conditions on the operator $A$ that ensure the existence and uniqueness of a (weak) solution $x \in C_{b}\cap \mathscr{F}$ for any function $f \in \mathscr{F}$. This problem is equivalent to the differential operator

$$\begin{equation*} \mathscr{L}=\mathscr{L}_{\mathscr{F}}=\frac{d}{dt}-A\colon D(\mathscr{L}) \subset \mathscr{F} \to \mathscr{F}=\mathscr{F}(\mathbb{R},\mathscr{H}) \end{equation*}$$

being continuously invertible. This is defined as follows. A function $x \in C_{b}\cap \mathscr{F}$ is considered to belong to the domain of definition $D(\mathscr{L}_{\mathscr{F}})$ of the operator if there exists a function $f \in \mathscr{F}$ such that $x$ is a weak solution of the differential equation (1.1), that is, equations (1.2) hold. The operator $\mathscr{L}_{\mathscr{F}}$ is said to be continuously invertible if its kernel $\operatorname{Ker}\mathscr{L}_{\mathscr{F}}$ is zero and the image $\operatorname{Im}\mathscr{L}_{\mathscr{F}}$ coincides with $\mathscr{F}$. Consequently, $\mathscr{L}_{\mathscr{F}}^{-1} \in \operatorname{End} \mathscr{F}$.

Definition 1.  An operator semigroup $T\colon \mathbb{R}_{+} \to \operatorname{End}\mathscr{H}$ is said to be hyperbolic or admitting exponential dichotomy if the spectrum $\sigma(T(1))$ of the operator $T(1)$ has the property

$$\begin{equation} \sigma(T(1)) \cap \mathbb{T}=\varnothing, \end{equation} \tag{ 1.3 }$$

where $\mathbb{T}=\{\lambda \in \mathbb{C}: |\lambda|=1\}$ is the unit circle.

The following result is contained in [2]–[5].

Theorem 1.  For the operator $\mathscr{L}\colon D(\mathscr{L})\subset\mathscr{F}\to\mathscr{F}$ to be continuously invertible it is necessary and sufficient that the semigroup $T$ be hyperbolic.

Theorem 2, which follows, was proved by Prüss (see [6], [7]) for operator semigroups of class $C_{0}$ (see also [8] and [9] for degenerate operator semigroups, where the role of the operator $A$ is taken by a linear relation [9]).

Theorem 2.  For a (strongly continuous) operator semigroup $T\colon \mathbb{R}_{+} \to \operatorname{End} \mathscr{H}$ to be hyperbolic, it is necessary and sufficient that the following two conditions hold:

$$\begin{equation} \sigma(A) \cap (i \mathbb{R})=\varnothing, \end{equation} \tag{ 1.4 }$$

$$\begin{equation} \sup_{\lambda \in \mathbb{R}}\|R(i\lambda,A)\| < \infty, \end{equation} \tag{ 1.5 }$$

where $R(\,\cdot\,,A)\colon \rho(A) \subset \mathbb{C} \to \mathscr{H}$ is the resolvent of the operator $A$ and $\rho(A)=\mathbb{C} \setminus \sigma(A)$ is its resolvent set.

When an operator semigroup $T\colon \mathbb{R}_{+} \to\operatorname{End} \mathscr{H}$ is hyperbolic, this gives rise to the decomposition

$$\begin{equation} \sigma(T(1))=\sigma_{int} \cup \sigma_{out}, \end{equation} \tag{ 1.6 }$$

where $\sigma_{int}=\{\lambda \in \sigma(T(1)):|\lambda| < 1\}$ and $\sigma_{out}=\{\lambda \in \sigma(T(1)):|\lambda| > 1\}$. We consider the Riesz projector (see [1]–[3])

$$\begin{equation} P_{int}=\frac{1}{2\pi i}\int_{\mathbb{T}}(T(1)-\lambda I)^{-1}\,d\lambda \end{equation} \tag{ 1.7 }$$

constructed with respect to the set $\sigma_{int}$ and put $P_{out}=I-P_{int}$, where $I$ is the identity operator in the algebra $\operatorname{End}\mathscr{H}$. Then the Hilbert space $\mathscr{H}$ can be represented as the direct sum of the closed linear subspace $\mathscr{H}_{int}=\operatorname{Im} P_{int}$ that is the image of the projector $P_{int}$ and the subspace $\mathscr{H}_{out}=\operatorname{Im} P_{out}$:

$$\begin{equation*} \mathscr{H}=\mathscr{H}_{int}\oplus \mathscr{H}_{out}. \end{equation*}$$

It follows from (1.7) that the projectors $P_{int}$, $P_{out}$ commute with the operators $T(t)$, $t \geqslant 0$, and, consequently, the subspaces $\mathscr{H}_{int}$, $\mathscr{H}_{out}$ are invariant under the operators $T(t)$, $t \geqslant 0$. Furthermore, we note that these subspaces are nonzero if $\sigma_{int} \ne\varnothing$, $\sigma_{out} \ne \varnothing$ simultaneously for the sets in the decomposition (1.6). We consider (for more details see [4] and [10]) the operator semigroup

$$\begin{equation*} T_{int}\colon \mathbb{R}_{+} \to \operatorname{End}\mathscr{H}_{int}, \qquad T_{int}(t)=T(t)|\mathscr{H}_{int},\quad t \geqslant 0, \end{equation*}$$

and the operator group $T_{out}\colon \mathbb{R} \to\operatorname{End}\mathscr{H}_{out}$ defined by the equations

$$\begin{equation*} T_{out}(t)=\begin{cases} T(t)|\mathscr{H}_{out},& t \geqslant 0, \\ (T(-t) \mid \mathscr{H}_{out})^{-1}, & t < 0. \end{cases} \end{equation*}$$

Theorem 3. (see [2]–[4]) If the semigroup $T$ is hyperbolic, then the operator $\mathscr{L}$ is continuously invertible and the inverse operator $\mathscr{L}^{-1} \in \operatorname{End} \mathscr{F}$ is defined by the formula

$$\begin{equation} (\mathscr{L}^{-1}f)(t)=(G \ast f)(t)=\int_{\mathbb{R}} G(t-s)f(s)\,ds,\qquad t \in \mathbb{R},\quad f \in \mathscr{F}. \end{equation} \tag{ 1.8 }$$

The Green's function $G=G_{A}\colon \mathbb{R} \to \operatorname{End} \mathscr{H}$ has the form

$$\begin{equation} G(t)=\begin{cases} T(t)P_{int}, & t \geqslant 0, \\ -T_{out}(t)P_{out}, & t < 0. \end{cases} \end{equation} \tag{ 1.9 }$$

For a hyperbolic semigroup $T$ it is possible to derive the existence of constants $M_{\pm}\geqslant 1$, $0 <\gamma_{\pm} \leqslant \infty$, such that the Green's function $G$ admits estimates of the form

$$\begin{equation} \|G(u)\|=\begin{cases} M_{+} e^{-\gamma_{+}u}, & u \geqslant 0, \\ M_{-} e^{\gamma_{-}u}, & u < 0. \end{cases} \end{equation} \tag{ 1.10 }$$

(See [2] and [4] for more details.)

Note that we set $\gamma_{-}=\infty$ if the semigroup $T$ is exponentially stable, that is,

$$\begin{equation*} \|T(t)\| \leqslant M_{+}e^{-\gamma_{+}t},\qquad t\geqslant 0, \end{equation*}$$

which is equivalent to the condition $\sigma_{out}=\varnothing$ (correspondingly we set $\gamma_{+}=\infty$ if $\sigma_{int}=\varnothing$).

The numbers $\gamma_{+}$, $\gamma_{-}$ in (1.10) are called the exponents of exponential dichotomy of the function $T$, and the two ordered pairs of numbers $(M_{-},\gamma_{-})$, $(M_{+},\gamma_{+})$ are called the parameters of exponential dichotomy. If the semigroup $T$ is exponentially stable, then we call the pair of numbers $(M_{+},\gamma_{+})$ the parameters of exponential stability of the semigroup $T$.

The representation (1.8) of the inverse operator of $\mathscr{L}$ (under the condition (1.3) that the semigroup $T$ is hyperbolic or under the conditions (1.4), (1.5)) implies that the norm of the operator $\mathscr{L}^{-1}\colon\mathscr{F} \to \mathscr{F}= \mathscr{F}(\mathbb{R},\mathscr{H})$ admits estimates of the form

$$\begin{equation} \|\mathscr{L}^{-1}\| \leqslant \|G\|_{1}=\int^{\infty}_{-\infty} \|G(\tau)\|\,d\tau \leqslant\frac{M_{-}}{\gamma_{-}}+ \frac{M_{+}}{\gamma_{+}}\,. \end{equation} \tag{ 1.11 }$$

The estimates (1.11) allows us to obtain a condition for the solvability in the space $C_{b}(\mathbb{R},\mathscr{H})$ of a nonlinear equation of the form

$$\begin{equation} \dot{x}(t)=Ax(t)+f(t,x(t)),\qquad t \in \mathbb{R}, \end{equation} \tag{ 1.12 }$$

where $f\colon\mathbb{R} \times \mathscr{H} \to \mathscr{H}$ is a continuous function (in the first argument) and is uniformly continuous with respect to the second argument in any ball in $\mathscr{H}$ and satisfies a Lipschitz condition in the second variable:

$$\begin{equation*} \|f(t,x)-f(t,y)\| \leqslant \ell\|x-y\|,\qquad x,y \in\mathscr{H}, \end{equation*}$$

where $\ell > 0$ is some constant. Then the following theorem holds.

Theorem 4.  Equation (1.12) has a weak solution $x_{*}\in C_{b}=C_{b}(\mathbb{R},\mathscr{H})$ if the condition $q=\| G\|_{1} \ell < 1$ holds. Furthermore,

$$\begin{equation*} \|x_{*}\|_{\infty} \leqslant \frac{\|G\|_{1}}{1-\|G\|_{1}\ell}\|f_{0}\|_{\infty}, \end{equation*}$$

where $f_{0}(t)=f(t,0)$, $t \in \mathbb{R}$.

Proof.  The proof of an analogue of Theorem 4 can be found in [11]; a definition of a weak solution of a nonlinear equation can also be found there. This proof is based on an application of the contraction mapping principle [12] to the integral equation

$$\begin{equation*} x(t)=(\Phi(x))(t)=\int_{\mathbb{R}}G(t-s)f(s,x(s))\,ds,\qquad t \in \mathbb{R}, \end{equation*}$$

where $\Phi\colon C_{b} \to C_{b}$ is a contraction map with contraction constant $\|\mathscr{L}^{-1}\|\ell \leqslant \|G\|_{1} \ell < 1$.

Thus, Theorem 4 shows that it is important to obtain an estimate for $\|G\|_{1}$. In this paper we use the following quantities for our estimates (see also the formulae in Theorem 8):

$$\begin{equation} \gamma(A)=\sup_{\lambda \in \mathbb{R}}\|R(i\lambda,A)\|, \end{equation} \tag{ 1.13 }$$

$$\begin{equation} \nu(A)=\sup_{\|x\|\leqslant 1}\frac{1}{2\pi}\int_{\mathbb{R}} \|R(it,A)x\|^{2}\,dt, \end{equation} \tag{ 1.14 }$$

$$\begin{equation} \nu(A^{*})=\sup_{\|x\|\leqslant 1}\frac{1}{2\pi}\int_{\mathbb{R}} \|R(it,A^{*})x\|^{2}\,dt, \end{equation} \tag{ 1.15 }$$

$$\begin{equation} k_{c}(T)=\sup_{0\leqslant t \leqslant c}\|T(t)\|,\qquad c > 0, \end{equation} \tag{ 1.16 }$$

$$\begin{equation} \beta(A)=\sup_{\mu \in \Theta(A)}\operatorname{Re}\mu, \end{equation} \tag{ 1.17 }$$

where $\Theta(A)=\{(Ax,x); \ \|x\|\leqslant 1,\ x \in D(A)\}$ is the numerical range of the operator $A$ (see [13]). Thus, in the estimates (1.13)–(1.15) we use quantities constructed from the resolvent of the operator $A$.

That $\gamma(A)$ is finite follows from Theorem 2. Its importance is that it gives the equality $\|\mathscr{L}^{-1}\|=\gamma(A)$ for the norm of the inverse operator of the differential operator $\mathscr{L}=\frac{d}{dt}-A\colon D(\mathscr{L}) \subset L^{2}(\mathbb{R},\mathscr{H}) \to L^{2}(\mathbb{R},\mathscr{H})$. This equality was proved in Lemma 4 of [4] using Parseval's equation. The quantity $\gamma(A)$ was used in [14], [4], [15] to estimate the norms of the inverse operators of differential operators. In [16] this quantity was called the frequency characteristic of the operator $A$.

The quantity $\nu(A)$ was used in [14] for $A \in \operatorname{End} \mathscr{H}$, where $\mathscr{H}$ is a finite- dimensional space. It was called the integral criterion of the quality of dichotomy. The estimate $\gamma(A) \leqslant 2\nu(A)$ was established in [10].

We also point out (without proof) the estimate $\gamma(A) \leqslant (\nu(A)\nu(A^{*}))^{1/2}$.

The quantity $\beta(A)$ defined by equation (1.17) with respect to the numerical range $\Theta(A)$ of the operator $A$ is important in connection with the following theorem (see [17], Theorem I.4.2 and §I.4.4).

Theorem 5.  For an operator $A\colon D(A) \!\subset\! \mathscr{H} \!\to\! \mathscr{H}$ with domain of definition $D(A)$, which is dense in $\mathscr{H}$, to be the infinitesimal operator of an operator semigroup $T\colon \mathbb{R}_{+} \to \operatorname{End}\mathscr{H}$ satisfying the estimate

$$\begin{equation} \|T(t)\| \leqslant e^{wt},\qquad t \geqslant 0, \end{equation} \tag{ 1.18 }$$

where $w \in \mathbb{R}$, it is necessary and sufficient that the following conditions hold:

$$\begin{equation} \operatorname{Re}(Ax,x) \leqslant w(x,x),\qquad x \in D(A), \end{equation} \tag{ 1.19 }$$

$$\begin{equation} \operatorname{Re}(A^{*}y,y) \leqslant w(y,y),\qquad y \in D(A^{*}). \end{equation} \tag{ 1.20 }$$

Thus, if (1.18) holds, this is equivalent to both the conditions $\beta(A) \leqslant w$ and $\beta(A^{*}) \leqslant w$ holding, that is, the numerical ranges $\Theta(A)$ and $\Theta(A^{*})$ of the operators $A$ and $A^{*}$ are contained in the half-plane $\mathbb{C}_{w}=\{\lambda \in \mathbb{C}: \operatorname{Re}\lambda \leqslant w\}$ (conditions (1.19), (1.20) hold). But a $w \in \mathbb{R}$ such that the estimate (1.18) is true does not exist for every semigroup $T\colon \mathbb{R}_{+} \to\operatorname{End}\mathscr{H}$. In §4 we present an example of an operator semigroup whose infinitesimal operator and its adjoint operator have numerical ranges that coincide with the entire complex plane.

The next assumption on the infinitesimal operator $A$ of an operator group $T\colon \mathbb{R}_{+} \to \operatorname{End}\mathscr{H}$ is used in the hypotheses of the following theorem.

Assumption 1.  The quantities

$$\begin{equation*} \beta_{\min}(A)=-\beta(-A)=\inf\{\operatorname{Re}\lambda: \lambda \in \Theta(A)\}, \qquad \beta_{\max}(A)=\beta(A) \end{equation*}$$

are finite, and $\beta_{\min}(A) < 0$, $\beta_{\max}(A) >0$.

Remark 1.  The conditions of Assumption 1 hold for an operator

$$\begin{equation*} A\colon D(A) \subset \mathscr{H} \to \mathscr{H}, \qquad\overline{D(A)}=\mathscr{H}, \end{equation*}$$

if $D(A^{*})=D(A)$ and the operator $2^{-1}(A+A^{*})\colon D(A) \subset \mathscr{H} \to \mathscr{H}$ admits a bounded extension to $\mathscr{H}$ to a (selfadjoint) operator $B \in \operatorname{End}\mathscr{H}$, whose spectrum $\sigma(B)$ satisfies the conditions $\sigma(B) \cap \mathbb{R}_{-} \ne \varnothing$ and $\sigma(B) \cap \mathbb{R}_{+} \ne \varnothing$, where ${\mathbb{R}_{-}=(-\infty,0]}$. In particular, these conditions are satisfied by the operator ${A=iA_{0}+ B}$, where $A_{0}\colon D(A_{0}) \subset \mathscr{H}\to \mathscr{H}$ is a selfadjoint operator and the spectrum of $B=B^{*} \in \operatorname{End}\mathscr{H}$ satisfies $\sigma(B)\cap\mathbb{R}_{\pm} \ne \varnothing$.

Theorem 6.  Let $T\colon \mathbb{R}\!\to\! \operatorname{End}\mathscr{H}$ be a hyperbolic operator group satisfying Assumption 1. Then the Green's function $G_{A}\colon \mathbb{R}\to \operatorname{End}\mathscr{H}$ admits an estimate of the form

$$\begin{equation} \begin{gathered}\| G_{A}(t)\| \leqslant \begin{cases} \bigl(2|\beta_{\min}(A)|\nu(A)\bigr)^{1/2} \exp\biggl\{-\dfrac{1}{2\nu(A)}t\biggr\}, & t \geqslant 0, \\ \\ \bigl(2\beta_{\max}(A) \nu(A)\bigr)^{1/2} \exp\biggl\{\dfrac{1}{2\nu(A)}t\biggr\}, & t < 0, \end{cases}\end{gathered} \end{equation} \tag{ 1.21 }$$

$$\begin{equation} \|G_{A}\|_{1} \leqslant 2\nu(A)\bigl(2\beta_{\max}(A) \nu(A)\bigr)^{1/2}+ \bigl(2|\beta_{\min}(A)| \nu(A)\bigr)^{1/2}. \end{equation} \tag{ 1.22 }$$

Corollary 1.  Under the conditions of Theorem 2 the norms of the projectors $P_{int}$, $P_{out}$ occurring in the definition of the Green's function (formula (1.9)) admit the estimate

$$\begin{equation*} \max\{\|P_{int}\|,\|P_{out}\|\} \leqslant \bigl(2\min\{|\beta_{\min}(A)|,\beta_{\max}(A)\}\nu(A)\bigr)^{1/2}. \end{equation*}$$

In proving Theorem 6 we make essential use of Theorem 6 in [10], in which a necessary and sufficient condition for operator groups to be hyperbolic was obtained. It was based on using Lyapunov's equation (see also [3], Theorem 4.40). The proof of Theorem 6 depends on constructing the indefinite metric defined by a selfadjoint operator $W \in \operatorname{End} \mathscr{H}$ in the Hilbert space $\mathscr{H}$; with respect to this the solution of the corresponding differential equation decays exponentially as $t \to \infty$. In [10] an example was constructed of a hyperbolic semigroup of compact operators for which the operator $W=W^{*}\in\operatorname{End}\mathscr{H}$ defined from Lyapunov's equation is not invertible, that is, it does not seem possible to obtain analogues of Theorem 6 for operator semigroups in this way. For a finite-dimensional space $\mathscr{H}$, analogues of Theorem 6 were obtained in [14], [18]–[22].

Suppose that $T\colon \mathbb{R}_{+}\to\operatorname{End}\mathscr{H}$ is an exponentially stable operator semigroup. Then (see [3], [10], and [23]) Lyapunov's equation

$$\begin{equation} A^{*}X+ XA=-I \end{equation} \tag{ 1.23 }$$

has a solution $W=W^{*} \in \operatorname{End} \mathscr{H}$ which is defined by the formula

$$\begin{equation} Wx=-\int^{\infty}_{0} T(t)^{*} T(t)x\,dt,\qquad x\in\mathscr{H}. \end{equation} \tag{ 1.24 }$$

Thus, $WD(A) \subset D(A^{*})$ and $A^{*}Wx+ WAx=-x$, $x \in D(A)$. Furthermore, the operator $W$ is negative-definite but not is necessarily continuously invertible.

Theorem 7.  Suppose that $T\colon \mathbb{R}_{+} \to \operatorname{End}\mathscr{H}$ is an exponentially stable operator semigroup and the operator $W$ is continuously invertible. Then the following estimates hold:

$$\begin{equation} \|T(t)\| \leqslant \sqrt{\|W\|\,\|W^{-1}\|}\,\exp\biggl\{-\frac{1}{2\|W\|}t\biggr\},\qquad t \geqslant 0. \end{equation} \tag{ 1.25 }$$

Note that for an exponentially stable operator group $T\colon \mathbb{R} \to \operatorname{End}\mathscr{H}$ the operator $W$ is invertible (see [10] and Theorem 17). It follows from Remark 2 (see §2) that $\|W\|=\nu(A)$.

In the case of a finite-dimensional space $\mathscr{H}$ the estimate (1.25) was given in [14], §9.3.

In §3 we describe a different method for obtaining an estimate for the Green's function based on using the methods of harmonic analysis.

Along with the quantities defined by formulae (1.13)–(1.17), new quantities $\nu_{\varepsilon}(A)$, $\nu_{\varepsilon}(A^{*})$ are introduced into consideration in Theorem 8. They are defined for an arbitrary linear operator $A\colon D(A) \subset \mathscr{H}\to \mathscr{H}$ that has non-empty resolvent set $\rho(A)$ and for which the following quantities are finite:

$$\begin{equation*} s(A)=\sup\{\operatorname{Re}\lambda : \lambda \in\sigma(A)\}, \qquad s_{0}(A)=\inf\Bigl\{a > s(A) : \sup_{\operatorname{Re}\lambda \geqslant a} \|R(\lambda,A)\| < \infty\Bigr\}. \end{equation*}$$

The quantity $s(A)$ is called the spectral bound of the operator $A$ (see [7], [23]), and the quantity $s_{0}(A)$ the abscissa of uniform boundedness of the resolvent of the operator $A$.

For an arbitrary (strongly continuous) operator semigroup $T\colon \mathbb{R}_{+} \to \operatorname{End}\mathscr{H}$, the quantity

$$\begin{equation*} w_{0}(T)=\inf\Bigl\{w \in \mathbb{R}:\sup_{t \geqslant 0}e^{-wt} \|T(t)\|<\infty\Bigr\} \end{equation*}$$

is called the type of the semigroup $T$ or the growth constant of the semigroup $T$ (see [7], [23]).

Note that $s(A) \leqslant s_{0}(A) \leqslant w_{0}(T)$ for the infinitesimal operator $A$ of the semigroup $T$. Furthermore, $s_{0}(A)=w_{0}(T)$ by Theorem 2. For operator semigroups in a Banach space these quantities may be different [7].

Theorem 8.  An operator $A\colon D(A) \subset \mathscr{H}\to \mathscr{H}$ with $\overline{D(A)}=\mathscr{H}$ is the infinitesimal operator of an exponentially stable operator semigroup $T\colon \mathbb{R}_{+} \to \operatorname{End}\mathscr{H}$ if and only if for some $\varepsilon > 0$ the following conditions hold:

• 1)  
  $s_{0}(A) < 0;$
• 2)  
  $\nu_{\varepsilon}(A)=\displaystyle{\sup_{a > 0}\, \sup_{\|x\|\leqslant 1}a\int^{\infty}_{-\infty}} \|R(a+s_{0}(A)+\varepsilon+ i\lambda,A)x\|^{2}\,d\lambda <\infty$;
• 3)  
  $\nu_{\varepsilon}(A^{*})=\displaystyle{\sup_{a > 0}\, \sup_{\|x\|\leqslant 1}a\int^{\infty}_{-\infty}} \|R(a+ s_{0}(A)+\varepsilon+i\lambda, A^{*})x\|^{2}\,d\lambda < \infty$.

When these conditions hold, the semigroup $T\colon \mathbb{R}_{+} \to \operatorname{End}\mathscr{H}$ admits the estimate

$$\begin{equation} \|T(t)\| \leqslant \frac{e}{2\pi}\sqrt{\nu_{\varepsilon}(A) \nu_{\varepsilon}(A^{*})}\,\exp\bigl\{(s_{0}(A)+\varepsilon)t\bigr\},\qquad t \geqslant 0. \end{equation} \tag{ 1.26 }$$

Furthermore, conditions 1)– 3) hold for any $\varepsilon > 0$.

Theorem 8 is essential to the proof of Theorem 18 in §3, where estimates of the Green's function for sectorial operators are obtained.

In view of the difficulty of obtaining estimates for $\nu_{\varepsilon}(A)$ and $\nu_{\varepsilon}(A^{*})$, the quantities $\gamma(A)$, $\nu(A)$, $\nu(A^{*})$, $k_{c}(T)$ defined by formulae (1.13)–(1.16) are used in the following theorems.

The estimates presented below in Theorems 9–11 are obtained for a hyperbolic operator semigroup $T\colon \mathbb{R}_{+} \to \operatorname{End}\mathscr{H}$ with infinitesimal operator $A$.

Theorem 9.  The Green's function $G_{A}\colon \mathbb{R} \to \operatorname{End}\mathscr{H}$ admits estimates of the form

$$\begin{equation} \|G_{A}(t)\| \leqslant\frac{1}{|t|}\, \frac{\sqrt{\nu(A)\nu(A^{*})}}{(1-\alpha\gamma(A))^{2}}\exp\{-\alpha|t|\},\qquad t \ne 0, \end{equation} \tag{ 1.27 }$$

for any number $\alpha > 0$ satisfying the condition $\alpha <{1}/{\gamma(A)}$. In particular (for $\alpha=(2\gamma(A))^{-1})$, the following estimates hold:

$$\begin{equation} \|G_{A}(t)\| \leqslant \frac{4}{|t|}\sqrt{\nu(A)\nu(A^{*})}\, \exp\biggl\{-\frac{|t|}{2\gamma(A)}\biggr\},\qquad t \ne 0. \end{equation} \tag{ 1.28 }$$

Corollary 2.  Suppose that $T$ is an exponentially stable operator semigroup. Then $w_{0}(T) \leqslant (\gamma(A))^{-1}$.

The next two theorems follows from Theorem 9.

Theorem 10.  Let $T\colon \mathbb{R}_{+} \to \operatorname{End}\mathscr{H}$ be an exponentially stable operator semigroup. Then it admits estimates of the form

$$\begin{equation} \|T(t)\| \leqslant \begin{cases} k_{c}(T),& 0 \leqslant t \leqslant c, \\ \dfrac{1}{t}\,\dfrac{\sqrt{\nu(A)\nu(A^{*})}}{(1-\alpha\gamma(A))^{2}}\,\exp\{-\alpha t\},& t > c, \end{cases} \end{equation} \tag{ 1.29 }$$

for any number $\alpha > 0$ satisfying the condition $\alpha < 1/\gamma(A)$ and for any $c > 0$.

In the following theorem, the quantity $\beta(A)$ defined by equation (1.17) is used in estimating the quantity $k_{c}(T)$. Furthermore, Theorem 5 is used.

Theorem 11.  If $\beta(A) < \infty$ and $T\colon \mathbb{R}_{+} \to \operatorname{End}\mathscr{H}$ is an exponentially stable operator semigroup, then

$$\begin{equation} \|T(t)\| \leqslant \begin{cases} e^{\beta(A)t}, & 0 \leqslant t \leqslant c, \\ \dfrac{4}{t}\sqrt{\nu(A)\nu(A^{*})}\,\exp\biggl\{-\dfrac{1}{2\gamma(A)}t\biggr\}, & t > c, \end{cases} \end{equation} \tag{ 1.30 }$$

for any number $c > 0$. Furthermore,

$$\begin{equation*} \|\mathscr{L}^{-1}\| \leqslant \|G_{A}\|_{1} \leqslant \frac{e^{\beta(A)c}-1}{\beta(A)}+ \frac{8}{c}\sqrt{\nu(A)\nu(A^{*})}\,\gamma(A). \end{equation*}$$

Suppose that

$$\begin{equation*} \mathscr{L}=\frac{d}{dt}-Q\colon D(\mathscr{L})= W^{1}_{2}(\mathbb{R},H) \subset L^{2}(\mathbb{R},H) \to L^{2}(\mathbb{R},H),(\mathbb{R},H), \end{equation*}$$

where $W^{2,1}(\mathbb{R},H)=\{x \in L^{2}(\mathbb{R}, H): x$ is absolutely continuous and $\dot{x} \in L^{2}(\mathbb{R},H)\}$ is a Sobolev space and $Q \in C_{b}(\mathbb{R},\operatorname{End} H)$, is a continuously invertible operator. It was proved in [15] that the operator $\mathscr{L}$ is continuously invertible in $C_{b}(\mathbb{R},\operatorname{End}H)$ and that

$$\begin{equation*} \|\mathscr{L}^{-1}\|_{\infty} \leqslant 4\bigl(\|\mathscr{L}^{-1}\|_{2}+ \|Q\|_{\infty}\| \mathscr{L}^{-1}\|_{2}^{2}\bigr), \end{equation*}$$

where $\|\mathscr{L}^{-1}\|_{\infty}$ and $\|\mathscr{L}^{-1}\|_{2}$ are the norms of the operator $\mathscr{L}^{-1}$ in $C_{b}$ and $L^{2}$, respectively. In the proof in [15], completely different methods were used (these were later used in the theory of wavelets [24]).

The results in §5 are connected with estimates for the Green's function constructed for a hyperbolic linear relation. We present the basic ideas in the theory of linear relations we use here; these are described in detail in [9], [25]–[27] (also see [28] and [29]).

Let $\mathscr{X}$ be a complex Banach space. Any linear subspace $\mathscr{A}$ in the Cartesian product $\mathscr{X}\times\mathscr{X}$ is called a linear relation on the Banach space $\mathscr{X}$. In what follows, any linear operator $A\colon D(A)\subset \mathscr{X}\to \mathscr{X}$ is identified with the linear relation

$$\begin{equation*} \mathscr{A}=\bigl\{(x,Ax)\in \mathscr{X}\times\mathscr{X}:x \in D(A)\bigr\}, \end{equation*}$$

which is the graph of the operator $A$. Thus, we have the inclusion

$$\begin{equation*} \operatorname{End}\mathscr{X}\subset LO(\mathscr{X})\subset LR(\mathscr{X}), \end{equation*}$$

where $LO(\mathscr{X})$ is the set of linear operators acting in $\mathscr{X}$ and $LR(\mathscr{X})$ is the set of linear relations on $\mathscr{X}$. The set of linear relations that are closed in $\mathscr{X}\times\mathscr{X}$ is denoted by $LRC(\mathscr{X})$.

The subspace

$$\begin{equation*} D(\mathscr{A})=\{x \in \mathscr{X}: \ \text{there exists} \ y \in \mathscr{X} \ \text{such that} \ (x, y) \in \mathscr{A}\} \end{equation*}$$

is called the domain of definition of the relation $\mathscr{A} \in LR(\mathscr{X})$. Let $\mathscr{A}x$, where ${x \in D(A)}$, denote the set $\{y \in \mathscr{X}: (x,y) \in \mathscr{A}\}$. The subspaces $\operatorname{Ker}\mathscr{A}=\{{x \in D(\mathscr{A})}: (x,0) \in \mathscr{A}\}$ and $\operatorname{Im}\mathscr{A}= \bigcup_{x \in D(\mathscr{A})}\mathscr{A}x$ are called the kernel and image of $\mathscr{A} \in LR(\mathscr{X})$, respectively.

The sum of two relations $\mathscr{A,B} \in LR(\mathscr{X})$ is defined to be a linear subspace of $\mathscr{X}\times\mathscr{X}$ of the form

$$\begin{equation*} \mathscr{A}+ \mathscr{B}=\bigl\{(x,y) \in \mathscr{X}\times\mathscr{X}: x \in D(\mathscr{A})\cap D(\mathscr{B}), \ y \in \mathscr{A}x+\mathscr{B}x\bigr\}, \end{equation*}$$

where $\mathscr{A}x+ \mathscr{B}x$ is the algebraic sum of two subsets $\mathscr{A}x$, $\mathscr{B}x$. The product of linear relations $\mathscr{A,B} \in LR(\mathscr{X})$ is defined to be a linear subspace of $\mathscr{X}\times\mathscr{X}$ of the form

$$\begin{equation*} \begin{aligned} \mathscr{AB} &=\bigl\{(x,z) \in \mathscr{X}\times\mathscr{X}: \\ &\qquad \text{there exists } y \in D(\mathscr{A}) \ \text{such that} \ (x,y) \,{\in}\, \mathscr{B},\, (y, z) \,{\in}\, \mathscr{A}\bigr\}. \end{aligned} \end{equation*}$$

If $\mathscr{A} \in LR(\mathscr{X})$, then the inverse relation $\mathscr{A}^{-1} \in LR(\mathscr{X})$ is defined by the equation $\mathscr{A}^{-1}=\{(y,x) \in \mathscr{X}\times\mathscr{X}: (x,y) \in \mathscr{A}\}$. Clearly, $D(\mathscr{A}^{-1})=\operatorname{Im}\mathscr{A}$ and ${\operatorname{Im}\mathscr{A}^{-1}=D(\mathscr{A})}$.

A relation $\mathscr{A}$ in $LRC(\mathscr{X})$ is said to be continuously invertible if $\mathscr{A}^{-1} \in \operatorname{End}\mathscr{X}$, that is, $\operatorname{Ker}\mathscr{A}=\{0\}$ ($\mathscr{A}$ is injective) and $\operatorname{Im}\mathscr{A}=\mathscr{X}$ ($\mathscr{A}$ is surjective). The resolvent set of a relation $\mathscr{A}$ is defined to be the set $\rho(\mathscr{A})$ of all $\lambda \in \mathbb{C}$ such that $(\mathscr{A}-\lambda I)^{-1} \in \operatorname{End} \mathscr{X}$, that is, the relation $\mathscr{A}-\lambda I$ is continuously invertible. The spectrum of $\mathscr{A} \in LRC(\mathscr{X})$ is defined to be the set $\sigma(\mathscr{A})=\mathbb{C} \setminus \rho(\mathscr{A})$.

It should be noted that the set $\rho(\mathscr{A})$ is open, and the spectrum $\sigma(\mathscr{A})$ of ${A \in LRC (\mathscr{X})}$ is closed. The function

$$\begin{equation*} R(\,\cdot\,,\mathscr{A})\colon \rho(\mathscr{A}) \to \operatorname{End}\mathscr{X},\qquad R(\lambda, \mathscr{A})=(\mathscr{A}-\lambda I)^{-1},\quad \lambda\in\rho(\mathscr{A}), \end{equation*}$$

is called the resolvent of $\mathscr{A}\in LRC(\mathscr{X})$.

A closed linear subspace $\mathscr{X}_{0} \subset \mathscr{X}$ is said to be invariant for a relation ${\mathscr{A}\in LRC(\mathscr{X})}$ with non-empty $\rho(\mathscr{A})$ if it is invariant under all the operators $R(\lambda,\mathscr{A})$, $\lambda \in \rho(\mathscr{A})$. The restriction of a relation $\mathscr{A}$ to a subspace $\mathscr{X}_{0}$ is defined to be the relation $\mathscr{A}_{0} \in LRC(\mathscr{X}_{0})$ the resolvent of which is the restriction $R_{0}\colon\rho(\mathscr{A})\to\operatorname{End}\mathscr{X}_{0}$, $R_{0}(\lambda)=R(\lambda,\mathscr{A}) \mid \mathscr{X}_{0}$, $\lambda \in \rho(\mathscr{A})$, of the resolvent $R(\,\cdot\,,\mathscr{A})\colon \rho(\mathscr{A}) \to \operatorname{End}\mathscr{X}$ to the subspace $\mathscr{X}_{0}$; it is denoted by $\mathscr{A}_{0}=\mathscr{A} \mid \mathscr{X}_{0}$.

If $\mathscr{X}=\mathscr{X}_{0} \oplus \mathscr{X}_{1}$ is a direct sum of subspaces invariant under $\mathscr{A}\in LRC(\mathscr{X})$ and $\mathscr{A}_{0}=\mathscr{A} \mid\mathscr{X}_{0}$, $\mathscr{A}_{1}=\mathscr{A} \mid \mathscr{X}_{1}$, then we say that $\mathscr{A}$ is the direct sum of the relations $\mathscr{A}_{0}$ and $\mathscr{A}_{1}$ and write $\mathscr{A}=\mathscr{A}_{0} \oplus \mathscr{A}_{1}$. An operator $B \in \operatorname{End}\mathscr{X}$ is said to commute with a relation $\mathscr{A} \in LRC(\mathscr{X})$ if $(Bx,By)\in \mathscr{A}$ for any $(x, y) \in \mathscr{A}$.

Definition 2.  A relation $\mathscr{A} \in LRC(\mathscr{X})$ is said to be hyperbolic (with respect to the circle $\mathbb{T}$) if

$$\begin{equation} \sigma(\mathscr{A})\cap \mathbb{T}=\varnothing . \end{equation} \tag{ 1.31 }$$

If (1.31) holds for $\mathscr{A} \in LRC(\mathscr{X})$ then we have the representation

$$\begin{equation} \sigma(\mathscr{A})=\sigma_{int} \cup\sigma_{out}, \end{equation} \tag{ 1.32 }$$

where $\sigma_{int}=\{\lambda \in \sigma(\mathscr{A}):|\lambda| < 1\}$ and $\sigma_{out}=\{\lambda \in \sigma(\mathscr{A}):|\lambda| > 1\}$. The following result follows from [25] (see also Theorem 4.2 in [26] and Theorem 5.2.10 in [27]).

Theorem 12.  Let $\mathscr{A}$ be a hyperbolic linear relation on a space $\mathscr{X}$. Then the space $\mathscr{X}$ admits a representation in the form

$$\begin{equation} \mathscr{X}=\mathscr{X}_{int} \oplus \mathscr{X}_{out} \end{equation} \tag{ 1.33 }$$

of the direct sum of closed subspaces $\mathscr{X}_{int}$, $\mathscr{X}_{out}$ invariant under the linear relation $\mathscr{A}$, and $\mathscr{A}=\mathscr{A}_{0} \oplus \mathscr{A}_{1}$, where $\mathscr{A}_{0}=\mathscr{A} \mid \mathscr{X}_{int}$ and $\mathscr{A}_{1}= \mathscr{A} \mid \mathscr{X}_{out} \in LRC(\mathscr{X}_{out})$ have the following properties:

• 1)  
  $\mathscr{A}_{0} \in \operatorname{End}\mathscr{X}_{0}$, $\sigma(\mathscr{A}_{0})=\sigma_{int}$;
• 2)  
  $\mathscr{A}_{1}0=\mathscr{A}0$, $D(\mathscr{A})=\mathscr{X}_{int} \oplus D(\mathscr{A}_{1})$, $\sigma(\mathscr{A}_{1})=\sigma_{out}$.

We obtain the decomposition (1.33) of the space $\mathscr{X}$ by the means of Riesz projector

$$\begin{equation} P_{0}=-\frac{1}{2\pi i}\int_{\mathbb{T}}R(\lambda,\mathscr{A})\,d\lambda \in \operatorname{End}\mathscr{X}, \end{equation} \tag{ 1.34 }$$

which commutes with the relation $\mathscr{A}$. Furthermore, $\mathscr{X}_{int}\!=\!\operatorname{Im} P_{0}$ and ${\mathscr{X}_{out}\!=\!\operatorname{Im} P_{1}}$, where $P_{1}=I-P_{0}$ is the complementary projector to $P_{0}$.

Let $\ell^{p}=\ell^{p}(\mathbb{Z},\mathscr{X})$, where $p \in [1,\infty]$, denote the Banach space of two-sided sequences $x\colon \mathbb{Z} \to \mathscr{X}$ such that the quantity

$$\begin{equation*} \|x\|_{p}=\biggl(\,\sum_{n \in \mathbb{Z}}\|x(n)\|^{p}\biggr)^{1/p},\qquad p \in [1,\infty),\quad \|x\|_{\infty}=\sup_{n \in \mathbb{Z}}\|x(n)\|,\quad p=\infty, \end{equation*}$$

is finite; this is defined to be the norm in the space $\ell^{p}$.

We consider the difference inclusion

$$\begin{equation} x(n) \in Ax(n-1)+f(n),\qquad n \in \mathbb{Z}, \end{equation} \tag{ 1.35 }$$

where $f \in \ell^{p}(\mathbb{Z},\mathscr{X})$ and $A \in LRC(\mathscr{X})$. By a solution of this inclusion we mean a two-sided sequence $x \in \ell^{p}(\mathbb{Z},\mathscr{X})$ satisfying (1.35) for every $n \in \mathbb{Z}$, that is, $(x(n-1),x(n)-f(n))\in A$, $n \in \mathbb{Z}$. We construct a linear relation $\mathscr{A} \in LRC(\ell^{p})$ on the Banach space $\ell^{p}(\mathbb{Z},\mathscr{X})$. This relation consists of pairs $(x,y) \in \ell^{p} \times \ell^{p}$ such that

$$\begin{equation*} (x(k-1),y(k)) \in A,\qquad k \in \mathbb{Z}. \end{equation*}$$

We point out that the closedness of $A$ implies the closedness of $\mathscr{A}$. Based on Theorem 12 and the results in [30] (where the case $A \in LO(\mathscr{X})$ was considered), Bichegkuev [31] obtained the following result.

Theorem 13.  The following conditions are equivalent:

• 1)  
  the difference inclusion (1.35) has a unique solution $x \in \ell^{p}$ for any sequence $f \in \ell^{p}$;
• 2)  
  the relation $A$ is hyperbolic;
• 3)  
  the relation $I-A$ is continuously invertible.

If one of these conditions holds, then the operator $\mathscr{A}^{-1}\in \operatorname{End}\ell^{p}$ admits a representation of the form

$$\begin{equation} (\mathscr{A}^{-1}f)(n)=(G \ast f)(n)=\sum_{m \in \mathbb{Z}}G(n-m)f(m),\qquad n \in\mathbb{Z},\quad f \in \ell^{p}, \end{equation} \tag{ 1.36 }$$

where the function $G\colon \mathbb{Z} \to \operatorname{End}\mathscr{X}$, which is called the Green's function for the difference inclusion (1.35), has the form

$$\begin{equation} G(k)=\begin{cases} \mathscr{A}^{k}_{0}P_{0}, & k \geqslant 0, \\ -\mathscr{A}^{k}_{1}P_{1}, & k \leqslant -1, \end{cases} \end{equation} \tag{ 1.37 }$$

where $\mathscr{A}_{0}$, $\mathscr{A}_{1}$, $P_{0}$, $P_{1}$ are defined in Theorem 12.

To estimate the Green's function $G$, we use the quantity

$$\begin{equation*} \gamma_{\mathbb{T}}(A)=\sup_{\gamma \in \mathbb{T}}\| R(\gamma,A)\|. \end{equation*}$$

It follows from formula (1.36) that the norm of the operator $\mathscr{A}^{-1} \in\operatorname{End} \ell^{p}$ admits the estimate

$$\begin{equation*} \|\mathscr{A}^{-1}\| \leqslant \sum_{k \in \mathbb{Z}}\|G(k)\| \end{equation*}$$

in any of the Banach spaces $\ell^{p}(\mathbb{Z},\mathscr{X})$, $p\in [1,\infty]$.

Theorem 14.  Suppose that $A \in LR(\mathscr{X})$ is a hyperbolic relation. Then the following estimate holds:

$$\begin{equation} \sum_{k \in \mathbb{Z}}\|G(k)\| \leqslant \begin{cases} 4\gamma_{\mathbb{T}}(A)^{2}-2\gamma_{\mathbb{T}}(A) &\textit{if } \ \sigma_{int} \ne \varnothing, \ \sigma_{out}=\varnothing, \\ 4\gamma_{\mathbb{T}}(A)^{2}+ 2\gamma_{\mathbb{T}}(A) &\textit{if } \ \sigma_{int}=\varnothing, \ \sigma_{out} \ne \varnothing, \\ 8\gamma_{\mathbb{T}}(A)^{2} &\textit{if } \ \sigma_{int} \ne \varnothing, \ \sigma_{out} \ne \varnothing. \end{cases} \end{equation} \tag{ 1.38 }$$

Estimate (1.38) can be used in questions concerning the solvability in $\ell^{p}(\mathbb{Z},\mathscr{X})$ of the nonlinear difference inclusion

$$\begin{equation} x(n) \in Ax(n-1)+ g(n-1,x(n-1)),\qquad n \in \mathbb{Z}, \end{equation} \tag{ 1.39 }$$

where $A$ is a hyperbolic linear relation on $\mathscr{X}$ and $g\colon \mathbb{Z} \times \mathscr{X} \to \mathscr{X}$ is a map with the following properties:

• 1)  
  $\sup_{n \in \mathbb{Z}}\|g(n,m) < \infty\|$ for any $m \in \mathbb{Z}$,
• 2)  
  $\sup_{n \in \mathbb{Z}}\|g(n,k)-g(m,\ell)\| \leqslant \ell|k-m|$, $k \in \mathbb{Z}$, for some $\ell > 0$.

Theorem 15.  The difference inclusion (1.39) is solvable in the space $\ell^{\infty}(\mathbb{Z},\mathscr{X})$ if

$$\begin{equation*} 8\gamma_{\mathbb{T}}(A)^{2}\ell < 1. \end{equation*}$$

In §6 we present examples of hyperbolic operator semigroups. The examples and results presented here are closely connected with the results in [32]–[37]. In §7 we discuss some of the results obtained.

Throughout this section, $T\colon \mathbb{R}_{+} \to \operatorname{End}\mathscr{H}$ is a strongly continuous operator semigroup. If it is hyperbolic, then we use the notation introduced in §1.

Let $A$ be the infinitesimal operator of the hyperbolic operator semigroup $T\colon \mathbb{R}_{+} \to \operatorname{End}\mathscr{H}$. In the Banach algebra $\operatorname{End}\mathscr{H}$ we consider Lyapunov's equation

$$\begin{equation} A^{*}X+ XA=F \in \operatorname{End}\mathscr{H}. \end{equation} \tag{ 2.1 }$$

With the left-hand side of this equation we associate the transformer (following M. Krein's terminology)

$$\begin{equation*} \mathscr{L}\colon D(\mathscr{L}) \subset \operatorname{End}\mathscr{H} \to\operatorname{End}\mathscr{H}, \qquad \mathscr{L} X=A^{*}X+XA, \end{equation*}$$

with domain of definition

$$\begin{equation*} \begin{aligned} D(\mathscr{L})&=\bigl\{X \in \operatorname{End}\mathscr{H}: Xx \in D(A^{*}) \ \text{for all} \ x \in D(A) \\ &\qquad\text{and the operator} \ A^{*}X+XA \colon D(A) \subset \mathscr{H} \to \mathscr{H} \ \text{is bounded on $D(A)$}\bigl\}. \end{aligned} \end{equation*}$$

If $X \in D(\mathscr{L})$, then, since $D(A)$ is dense in $\mathscr{H}$, the extension of the operator $A^{*}X+XA$ to the space $\mathscr{H}$ is well defined and is denoted by the same symbol $A^{*}X+XA$.

We introduce into consideration closed subspaces in the algebra $\operatorname{End}\mathscr{H}$ of the form

$$\begin{equation*} \begin{aligned} \operatorname{End}_{*}\mathscr{H}&=\{X \in \operatorname{End}\mathscr{H}: P^{*}_{int}XP_{out}=P^{*}_{out}XP_{int}=0\} \\ &=\{X \in \operatorname{End}\mathscr{H}: P^{*}_{int}XP_{int}+ P^{*}_{out}XP_{out}=X\}, \\ \operatorname{End}_{0}\mathscr{H}&=\{X \in \operatorname{End}\mathscr{H}: P_{int}XP_{out}^{*}=P_{out}XP_{int}^{*}=0\} \\ &=\{X \in \operatorname{End}\mathscr{H}: P_{int}XP_{int}^{*}+ P_{out}XP_{out}^{*}=X\}. \end{aligned} \end{equation*}$$

The following two theorems were obtained in [3] and [10]. They are essential to the proofs of Theorems 6 and 7.

Theorem 16.  (see [3]) Suppose that the semigroup $T\colon \mathbb{R}_{+} \to \operatorname{End}\mathscr{H}$ is hyperbolic. Then equation (2.1) is solvable for any operator $F$ in the subspace $\operatorname{End}_{*}\mathscr{H}$, and there exists a unique solution $W$ belonging to the subspace $\operatorname{End}_{*}\mathscr{H}$. This solution can be represented by the formula

$$\begin{equation} \begin{aligned} Wx&=-\int^{\infty}_{0} G_{A}(t)^{*}FG_{A}(t)x\,dt+ \int^{0}_{\infty}G_{A}(t)^{*}FG_{A}(t)x\,dt \\ &=-\int^{\infty}_{0} \ T(t)^{*}P^{*}_{int}FT(t)P_{int}x\,dt+ \int^{0}_{\infty}T(t)^{*}P^{*}_{out}FT(t)P_{out}x\,dt,\qquad x \in\mathscr{H}. \end{aligned} \end{equation} \tag{ 2.2 }$$

In particular, if $F=-P^{*}_{int}P_{int}-P^{*}_{out}P_{out}$, then the operator $W$ has the form

$$\begin{equation} Wx=\int^{\infty}_{0}T(t)^{*}P_{int}^{*}T(t)P_{int}x\,dt- \int^{0}_{-\infty}T(t)^{*}P^{*}_{out}T(t)P_{out}x\,dt,\qquad x \in\mathscr{H}. \end{equation} \tag{ 2.3 }$$

Theorem 17.  (see [10]) For a (strongly continuous) operator group ${T\!\colon\! \mathbb{R}_{+} \!\!\to\! \operatorname{End}\!\mathscr{H}}$ to be hyperbolic, it is necessary and sufficient that there exist uniformly negative- definite operators $F \ll 0$, $F_{*} \ll 0$ in the algebra $\operatorname{End}\mathscr{H}$ such that the equations (2.1) and

$$\begin{equation} AX+XA^{*}=F_{*} \end{equation} \tag{ 2.4 }$$

have continuously invertible selfadjoint solutions $W \in \operatorname{End}_{*}\mathscr{H}$ and $W_{*} \in \operatorname{End}_{0}\mathscr{H}$, respectively. The solution $W$ is defined by (2.2), and the operator $W_{*}$ has the form

$$\begin{equation*} W_{*}x=-\int^{\infty}_{0}G_{A}(t)F_{*}G_{A}(t)^{*}x\,dt +\int^{0}_{\infty}G_{A}(t)F_{*}G_{A}(t)^{*}x\,dt,\qquad x\in \mathscr{H}. \end{equation*}$$

Remark 2.  It was established in [10] that the Fourier transform $\widehat{G}_{A}\colon \mathbb{R}\to \operatorname{End}\mathscr{H}$ of the Green's function of the hyperbolic operator semigroup $T\colon \mathbb{R}_{+} \to \operatorname{End}\mathscr{H}$ coincides with the restriction of the resolvent of the operator $A$ to the imaginary axis $i\mathbb{R}$, that is, the equations

$$\begin{equation} \widehat{G}_{A}(\lambda)x=\int_{\mathbb{R}}G_{A}(t)xe^{-i\lambda t}\,dt= R(i\lambda,A)x,\qquad \lambda \in \mathbb{R}, \end{equation} \tag{ 2.5 }$$

hold. Therefore it follows from formulae (2.2)–(2.5) (using Plancherel's identity) that the equations

$$\begin{equation*} (\widetilde{W}x,x)=\int_{\mathbb{R}}\|G_{A}(t)x\|^{2}\,dt= \frac{1}{2\pi}\int_{\mathbb{R}}\|R(i\lambda, A)x\|^{2}\,dt,\qquad x \in \mathscr{H}, \end{equation*}$$

hold if $F=P^{*}_{int}P_{int}-P^{*}_{out}P_{out}$ and $\widetilde{W}=W_{+}+ W_{-}$, where

$$\begin{equation*} W_{+}x=\int^{\infty}_{0}G_{A}(t)^{*}G_{A}(t)x\,dt,\qquad W_{-}x=\int^{0}_{\infty}G_{A}(t)^{*}G_{A}(t)x\,dt. \end{equation*}$$

Consequently,

$$\begin{equation} \|\widetilde{W}\|=\sup_{\|x\|\leqslant 1}|(\widetilde{W}x,x)|= \sup_{\|x\|\leqslant 1}\frac{1}{2\pi}\int_{\mathbb{R}} \|R(i\lambda,A)x\|^{2}\,d\lambda=\nu(A). \end{equation} \tag{ 2.6 }$$

Let $T\colon \mathbb{R}_{+} \to \operatorname{End}\mathscr{H}$ be a (strongly continuous) operator group. Along with the group $T$, we consider the operator group $\widetilde{T}\colon \mathbb{R}_{+} \to\operatorname{End}\mathscr{H}$, $\widetilde{T}(t)=T(-t)$, $t \in \mathbb{R}$. Its infinitesimal operator is the operator $-A$.

In proving Theorem 6 we use the following lemma.

Lemma 1.  The following estimates hold:

$$\begin{equation} e^{t\beta_{\min}(A)}\|x\| \leqslant \|T(t)x\| \leqslant e^{t\beta_{\max}(A)}\|x\|,\qquad t \geqslant 0, \end{equation} \tag{ 2.7 }$$

$$\begin{equation} e^{t\beta_{\max}(A)}\|x\| \leqslant \|T(t)x\| \leqslant e^{-t\beta_{\min}(A)}\|x\|,\qquad t \leqslant 0,\quad x \in \mathscr{H}. \end{equation} \tag{ 2.8 }$$

Proof.  In (2.7) the right-hand inequalities follow from Theorem 5, where $w=\beta_{\max}(A)$. Since $\beta_{\max}(-A)=-\beta_{\min}(A)$, we have

$$\begin{equation*} \|x\|=\|T(-t)T(t)x\| \leqslant \|T(-t)\|\,\|T(t)x\| \leqslant e^{-t\beta_{\min}(A)}\|T(t)x\|,\qquad t \geqslant 0. \end{equation*}$$

Consequently, the left-hand side of (2.7) also holds. The estimates (2.8) follow from (2.7) applied to the operator semigroup $\widetilde{T}$. Furthermore, we need to take the equation $\beta_{\min}(-A)=-\beta_{\max}(A)$ into account. The lemma is proved.

Proof  of Theorem 6 We consider the operators $W_{+}$, $W_{-}$, $\widetilde{W}$ in Remark 2 and take account of the fact that $\|\widetilde{W}\|=\nu(A)$ (see (2.6)). Since the operators $W_{+}$, $W_{-}$ are uniformly positive, we have

$$\begin{equation} (W_{\pm}x,x) \leqslant (\widetilde{W}x,x) \leqslant \|\widetilde{W}\|\, \|x\|^{2}=\nu(A)\|x\|^{2}, \qquad x \in \mathscr{H}. \end{equation} \tag{ 2.9 }$$

Lemma 1 implies the estimates

$$\begin{equation*} \begin{aligned} (W_{+}x,x)&=\int^{\infty}_{0}(T(\tau)P_{int}x,T(\tau)P_{int}x)\,d\tau =\int^{\infty}_{0}\|(T(\tau)P_{int}x\|^{2}\,d\tau \\ &\geqslant \int^{\infty}_{0} \ e^{2t\beta_{\min}(A)\tau}\|P_{int}x\|^{2}\,d\tau =\frac{1}{2|\beta_{\min}(A)|}\| P_{int}x\|^{2}. \end{aligned} \end{equation*}$$

Consequently, we have the estimates

$$\begin{equation} |(W_{+}x, x)| \geqslant \frac{1}{2|\beta_{\min}(A)|}\|P_{int} x\|^{2},\qquad x \in \mathscr{H}. \end{equation} \tag{ 2.10 }$$

Let $x_{0}$ be an arbitrary vector in $D(A)$. Consider the function $u\colon \mathbb{R}_{+} \to \mathscr{H}$, ${u(t)=G(t) x_{0}}$, $t \geqslant 0$. For $t \geqslant 0$ we obtain the equations

$$\begin{equation*} \begin{aligned} \frac{d}{dt}\bigl(W_{+}u(t),u(t)\bigr) &=\frac{d}{dt}\bigl(P^{*}_{int}W_{+} P_{int}u(t),u(t)\bigr) =\frac{d}{dt}\bigl(W_{+} P_{int}u(t),P_{int}u(t)\bigr) \\ &= \bigl(W_{+}P_{int}A u(t),P_{int} u(t)\bigr) + \bigl(W_{+}P_{int}u(t),P_{int}A u(t)\bigr) \\ &= \bigl(P^{*}_{int}W_{+}P_{int}Au(t),u(t)\bigr) +\bigl(P^{*}_{int} A^{*} W_{+}P_{int}u(t),u(t)\bigr) \\ &= \bigl((A^{*} W_{+}+W_{+}A)u(t),u(t)\bigr) =-\bigl(P^{*}_{int}P_{int}u(t),u(t)\bigr) \\ &= -\bigl(P_{int} u(t),P_{int} u(t)\bigr)=-\|P_{int} u(t)\|^{2}. \end{aligned} \end{equation*}$$

Using (2.9) with $x=P_{int}u(t)$ we deduce that

$$\begin{equation*} \begin{aligned} \bigl(W_{+} u(t),u(t)\bigr) &=\bigl(W_{+}P_{int} u(t),P_{int} u(t)\bigr) \\ &\leqslant \|\widetilde{W}\|\,\|P_{int} u(t)\|^{2}=\nu(A)\|P_{int} u(t)\|^{2}, \qquad t \geqslant 0. \end{aligned} \end{equation*}$$

Thus,

$$\begin{equation*} \frac{d}{dt}\bigl(W_{+}u(t),u(t)\bigr) \leqslant -\frac{1}{\nu(A)}\bigl(W_{+}u(t),u(t)\bigr),\qquad t \geqslant 0, \end{equation*}$$

and, consequently,

$$\begin{equation} \bigl(W_{+} u(t),u(t)\bigr) \leqslant\exp\biggl\{-\frac{1}{\nu(A)}t\biggr\} \bigl(W_{+} u(0),u(0)\bigr)=\exp\biggl\{-\frac{1}{\nu(A)}t\biggr\}(W_{+} x_{0},x_{0}). \end{equation} \tag{ 2.11 }$$

Using the estimates (2.9), (2.10) and (2.11) we obtain

$$\begin{equation*} \begin{aligned} & \frac{1}{2|\beta_{\min}(A)|}\|P_{int} u(t)\|^{2} \leqslant \bigl(W_{+} u(t),u(t)\bigr) \leqslant \exp\biggl\{-\frac{1}{\nu(A)}t\biggr\}(W_{+} x_{0},x_{0}) \\ &\qquad\leqslant \exp\biggl\{-\frac{1}{\nu(A)}t\biggr\}(\widetilde{W} x_{0},x_{0}) \leqslant \nu(A) \exp\biggl\{-\frac{1}{\nu(A)}t\biggr\}\|x_{0}\|^{2},\qquad t \geqslant 0. \end{aligned} \end{equation*}$$

Since $u(t)=P_{int} T(t)x_{0}=G_{A}(t) x_{0}$, $t \geqslant 0$, and $\overline{D(A)}=\mathscr{H}$, it follows that

$$\begin{equation*} \|G_{A}(t)x\|^{2} \leqslant 2|\beta_{\min}(A)|\nu(A) \exp\biggl\{-\frac{1}{\nu(A)}t\biggr\}\|x\|,\qquad t \geqslant 0, \end{equation*}$$

for any vector $x$ in $\mathscr{H}$. Thus,

$$\begin{equation} \|G_{A}(t)x\|\leqslant \bigl(2|\beta_{\min}(A)|\nu(A)\bigr)^{1/2} \exp\biggl\{-\frac{1}{\nu(A)}t\biggr\},\qquad t \geqslant 0. \end{equation} \tag{ 2.12 }$$

The estimate for $\|G_{A}(t)\|$ for $t < 0$ is obtained from the preceding one if we consider the operator group $\widetilde{T}(t)=T(-t)$, $t \in \mathbb{R}$, for $t \geqslant 0$. Since the operator $(-A)$ is a generator, it follows that $G_{(-A)}(t)=T(-t)P_{out}$ for $t > 0$. Since $\beta_{\min}(-A)=-\beta_{\max}(A)$, for $t < 0$ we obtain that

$$\begin{equation} \|G_{A}(t)\|=\|G_{-A}(-t)\| \leqslant\bigl (2 \beta_{\max}(A)\nu(A)\bigr)^{1/2} \exp\biggl\{-\frac{1}{\nu(A)}t\biggr\},\qquad t <0. \end{equation} \tag{ 2.13 }$$

The estimates (2.12), (2.13) obtained above imply the estimates (1.21) and (1.22) that were to be proved. Theorem 6 is proved.

Proof  of Theorem 7 Let $W=W^{*}$ be the operator defined by (1.24) and satisfying Lyapunov's equation (1.23), let $x_{0}$ be an arbitrary vector in $D(A)$, and let ${x(t)=T(t)x_{0}}$, $t \geqslant 0$. Then

$$\begin{equation*} \begin{aligned} &\frac{d}{dt}\bigl(Wx(t),x(t)\bigr)=\bigl(W\dot{x}(t),x(t)\bigr)+\bigl(Wx(t),\dot{x}(t)\bigr) \\ &\qquad =\bigl(WAx(t),x(t)\bigr)+\bigl(W x(t),Ax(t)\bigr) \\ &\qquad=\bigl((WA+A^{*}W)x(t),x(t)\bigr)=-\bigl(x(t),x(t)\bigr)=-\|x(t)\|^{2},\qquad t \geqslant 0. \end{aligned} \end{equation*}$$

It follows from the inequalities

$$\begin{equation*} (W y,y) \leqslant \|W\|\,\|y\|^{2},\qquad y \in D(A), \end{equation*}$$

that

$$\begin{equation*} \frac{d}{dt}\bigl(Wx(t),x(t)\bigr)=-\|x(t)\|^{2} \leqslant -\frac{1}{\|W\|}\bigl(Wx(t),x(t)\bigr),\qquad t \geqslant 0. \end{equation*}$$

Therefore,

$$\begin{equation*} \bigl(Wx(t),x(t)\bigr)\leqslant \exp\biggl\{-\frac{1}{\|W\|}t\biggr\}(W x_{0},x_{0}) \leqslant \|W\|\exp\biggl\{-\frac{1}{\|W\|}t\biggr\}\|x_{0}\|^{2}. \end{equation*}$$

Since

$$\begin{equation*} \bigl(Wx(t),x(t)\bigr) \geqslant \dfrac{1}{\|W^{-1}\|}\|x(t)\|^{2}, \qquad t \geqslant 0, \end{equation*}$$

we have

$$\begin{equation} \|x(t)\|^{2}=\|T(t)x_{0}\|^{2} \leqslant\|W^{-1}\|\,\|W\| \exp\biggl\{-\frac{1}{\|W\|}t\biggr\}\|x_{0}\|,\qquad t \in \mathbb{R}_{+},\quad x_{0} \in D(A). \end{equation} \tag{ 2.14 }$$

Since the operator semigroup $T\colon \mathbb{R}_{+} \to \operatorname{End}\mathscr{H}$ is bounded on any finite interval $[0,r]$, $r > 0$, and $D(A)$ is dense in the space $\mathscr{H}$, it follows from (2.14) that

$$\begin{equation*} \|T(t)\|^{2} \leqslant \|W^{-1}\|\,\|W\|\exp\biggl\{-\frac{1}{\|W\|}t\biggr\},\qquad t\geqslant 0. \end{equation*}$$

This inequality gives (1.25). The theorem is proved.

Proof  of Theorem 8 Necessity. Let $\widetilde{T}\colon \mathbb{R}_{+}\to \operatorname{End}\mathscr{H}$ be a (strongly continuous) bounded operator semigroup with infinitesimal operator $\widetilde{A}\colon D(\widetilde{A}) \subset \mathscr{H} \to \mathscr{H}$. Then $s(\widetilde{A}) \leqslant s_{0}(\widetilde{A})=w_{0}(\widetilde{T}) \leqslant 0$.

For any number $a > 0$ the operator semigroup $\widetilde{T}_{a}(t)=e^{-at} \widetilde{T}(t)$, $t \geqslant 0$, is an exponentially stable operator semigroup with infinitesimal operator $\widetilde{A}-a I$. Then equation (2.5) implies that

$$\begin{equation*} \begin{aligned} \widetilde{T}(t)x&=e^{at}\widetilde{T}_{a}(t)x=-\frac{1}{2\pi} \int^{\infty}_{-\infty}e^{(a+i\lambda)t} R(-a+i\lambda,\widetilde{A})x\,d \lambda \\ &=\frac{e^{at}}{2\pi t}\int^{\infty}_{-\infty}e^{i\lambda t} R(-a+ i\lambda,\widetilde{A})^{2}x\,d\lambda,\qquad x \in \mathscr{H}, \quad a > 0. \end{aligned} \end{equation*}$$

From this representation for any $x,y \in \mathscr{H}$ we obtain the estimates

$$\begin{equation} \begin{aligned} |(\widetilde{T}(t)x,y)|&=\frac{e^{at}}{2\pi t}\biggl|\int^{\infty}_{-\infty} e^{i\lambda t}\bigl(R(-a+i\lambda,\widetilde{A})x, R(-a-i\lambda,\widetilde{A}^{\ast})y\bigr)\,d\lambda\biggr| \\ &\leqslant \frac{e^{at}}{2\pi t a}\sup_{\|x\| \leqslant 1} \biggl(a \int^{\infty}_{-\infty}\|R(-a+ i\lambda,\widetilde{A}x\|^{2}\, d\lambda)^{1/2}\biggr) \\ &\qquad\times \biggl(\sup_{\|y\|\leqslant 1}\biggl(a\int^{\infty}_{-\infty} \|R(-a+ i\lambda,\widetilde{A}^{\ast})y\|^{2}\,d\lambda\biggr)^{1/2}. \end{aligned} \end{equation} \tag{ 3.1 }$$

For any number $\varepsilon > 0$ the operator semigroup

$$\begin{equation*} \widetilde{T}_{\varepsilon}\colon \mathbb{R}_{+} \to \operatorname{End}\mathscr{H}, \quad \widetilde{T}_{\varepsilon}(t)=e^{(-s_{0}(A)-\varepsilon)t}T(t),\qquad t \geqslant 0, \end{equation*}$$

is bounded. Its infinitesimal operator is the operator $\widetilde{A}_{\varepsilon}=A -(s_{0}(A)+\varepsilon) I$. Applying (3.1) to the operator semigroup $\widetilde{T}_{\varepsilon}$, we find that for any $a > 0$

$$\begin{equation*} \begin{aligned} \|\widetilde{T}_{\varepsilon}(t)\|&=e^{(-s_{0}(A)-\varepsilon)t}\|T(t)\| \\ &\leqslant\frac{e^{at}}{2\pi ta}\sup_{\|x\| \leqslant 1} \biggl(a\int^{\infty}_{-\infty}\|R(a+s_{0}(A)+\varepsilon+i\lambda,A)x\|^{2}\, d\lambda\biggr)^{1/2} \\ &\qquad\times\sup_{\|x\| \leqslant 1}\biggl(a\int^{\infty}_{-\infty} \|R(a+s_{0}(A)+\varepsilon+ i\lambda,A)x\|^{2}\,d\lambda\biggr)^{1/2},\qquad t > 0. \end{aligned} \end{equation*}$$

Using these estimates with $a=t^{-1}$ we obtain

$$\begin{equation*} \|T_{\varepsilon}(t) \| \leqslant \frac{e}{2\pi} (\nu_{\varepsilon}(A)\nu_{\varepsilon}(A^{*}))^{1/2},\qquad t >0. \end{equation*}$$

Consequently,

$$\begin{equation} \begin{aligned} \|T(t)\|&=e^{(s_{0}(A)+\varepsilon)t}\|T_{\varepsilon}(t)\| \\ &\leqslant \frac{e}{2\pi}(\nu_{\varepsilon}(A)\nu_{\varepsilon}(A^{*}))^{1/2} e^{(s_{0}(A)+\varepsilon)t},\qquad t > 0, \end{aligned} \end{equation} \tag{ 3.2 }$$

for any $\varepsilon > 0$. Thus, (1.26) follows from (3.2).

The sufficiency of the conditions in the theorem follows from Gomilko's theorem applied to the operator semigroup $\widetilde{T}_{\varepsilon}\colon \mathbb{R}_{+} \to \operatorname{End}\mathscr{H}$ (see [38]). Namely, if conditions 1)–3) of Theorem 8 hold, then the semigroup $\widetilde{T}_{\varepsilon}$ is bounded for any $\varepsilon > 0$. Therefore the semigroup $T(t)=e^{(s_{0}(A)+\varepsilon)t} T_{\varepsilon}(t)$, $t \geqslant 0$, is exponentially stable if we set $\varepsilon=-1/2 s_{0}(A)$. Theorem 8 is proved.

Theorem 8 can be used to estimate the parameter of exponential stability of analytic operator semigroups. Their infinitesimal operators are sectorial operators (see [7], [39]).

An operator $A\colon D(A) \subset \mathscr{H} \to \mathscr{H}$ is said to be sectorial with angle $\Theta \in (\pi/2,\pi)$ if for some $b \in \mathbb{R}$ the sector

$$\begin{equation*} \Omega=\Omega(a, \Theta)=\{\lambda \in \mathbb{C}: \arg(\lambda-a) < \Theta, \ \lambda \ne a\} \end{equation*}$$

is contained in the resolvent set $\rho(A)$ of the operator $A$ and for every $\delta \in (0,\Theta-\pi/2)$ there exists a number $M_{\delta} \geqslant 1$ such that

$$\begin{equation} \sup_{\lambda \in \Omega(b,\Theta-\delta)}\|(b-\lambda) R(\lambda, A)\|=M_{\delta} < \infty. \end{equation} \tag{ 3.3 }$$

Let $A\colon D(A) \subset \mathscr{H}\to \mathscr{H}$ be a sectorial operator whose spectrum is contained in the half-plane $\mathbb{C}_{-}=\{\lambda \in \mathbb{C}: \operatorname{Re}\lambda < 0\}$. It follows from the properties of sectorial operators (see [7], [39]) that the operator $A$ is the infinitesimal operator of some analytic operator semigroup $T\colon \mathbb{R}_{+} \to \operatorname{End}\mathscr{H}$, and the equations

$$\begin{equation*} s(A)=s_{0}(A)=w_{0}(T) \end{equation*}$$

hold.

We point out straight away that an operator $A$ being sectorial does not ensure that conditions (1.19), (1.20) in Theorem 5 hold. Moreover, in §4 we construct an example of a sectorial operator whose spectrum is negative. The operator has compact resolvent and its numerical range fills $\mathbb{C}$.

Since $s(A)<0$, the operator semigroup $T$ is exponentially stable. It follows from (3.3) that we can assume that $b<0$ in (3.3). Moreover, there exists a constant $M > 0$ such that

$$\begin{equation} \|R(\lambda,A)\| \leqslant \frac{M}{|\lambda-2^{-1}s(A)|} \end{equation} \tag{ 3.4 }$$

for all $\lambda \in \mathbb{C}$ contained in the half-plane

$$\begin{equation*} \mathbb{C}^{+}_{0}=\{\mu \in \mathbb{C}: \operatorname{Re}\mu > 2^{-1}s(A)=2^{-1}s_{0}(A)\}. \end{equation*}$$

Theorem 18.  Let $A\colon D(A) \subset \mathscr{H} \to \mathscr{H}$ be a sectorial operator such that $s(A) < 0$ whose resolvent satisfies the estimate (3.4) in the half-plane $\mathbb{C}^{+}_{0}$. Then a semigroup $T\colon \mathbb{R_{+}} \to \operatorname{End}\mathscr{H}$ whose infinitesimal operator is the operator $A$ admits the estimate

$$\begin{equation} \|T(t)\| \leqslant \frac{eM}{4} \exp\biggl\{\frac{-s(A)}{2}t\biggr\},\qquad t > 0, \end{equation} \tag{ 3.5 }$$

$$\begin{equation} \|G_{A}\|_{1}=\int^{\infty}_{0}\| T(\tau)\|\,d\tau\leqslant \frac{eM}{2s(A)}\,. \end{equation} \tag{ 3.6 }$$

Proof.  To obtain the estimates we use Theorem 8, setting $\varepsilon=2^{-1}s(A)$. For this $\varepsilon$ we obtain the following estimates for the quantities $\nu_{\varepsilon}(A)$ and $\nu_{\varepsilon}(A^{*})$:

$$\begin{equation*} \begin{aligned} \nu_{\varepsilon}(A)&=\sup_{a > 0}\,\sup_{\|x\| \leqslant 1} a\int^{\infty}_{-\infty}\biggl\|R\biggl(a+\frac{s(A)}{2}+ i\lambda,A\biggr)x\biggr\|^{2}\,d\lambda \\ &\leqslant \sup_{a > 0}\int^{\infty}_{-\infty}\frac{aM}{a^{2}+\lambda^{2}}\, d\lambda=\frac{\pi}{2}M. \end{aligned} \end{equation*}$$

Since

$$\begin{equation*} \biggl\|R\biggl(a+\dfrac{s(A)}{2}+i\lambda,A^{*}\biggr)\biggr\|= \biggl\|R\biggl(a+\dfrac{s(A)}{2}-i\lambda,A\biggr)\biggr\|, \end{equation*}$$

the same constant $\frac{\pi}{2}M$ estimates the quantity $\nu_{\varepsilon}(A^{*})$. The estimate (3.5) now follows from (1.26), and the estimate (3.6) from (3.5). The theorem is proved.

Proof  of Theorem 9 Using the representation of the function $\widehat{G}_{A}$ in the form (2.5), the equations $\frac{d}{d\lambda} R(i\lambda,A)=iR(i\lambda,A)^{2}$, $\lambda \in \mathbb{R}$, and the properties of the Fourier transform, we obtain the estimates

$$\begin{equation*} \begin{aligned} &\bigl|(itG_{A}(t)x,y)\bigr|=\frac{1}{2\pi}\biggl|\int_{\mathbb{R}} \bigl(R(i\lambda,A)^{2}x,y\bigr)e^{i\lambda t}\,d\lambda\biggr| \leqslant \frac{1}{2\pi}\int_{\mathbb{R}}\bigl|(R(i\lambda,A)^{2}x,y)\bigr|\,d\lambda \\ &\qquad=\frac{1}{2\pi}\int_{\mathbb{R}}\bigl|(R(i\lambda,A)x, R(-i\lambda,A^{*})y)\bigr|\,d\lambda \leqslant \frac{1}{2\pi}\!\int_{\mathbb{R}}\|R(i\lambda,A)x\|\, \|R (-i\lambda,A^{*}y)\|\,d\lambda \\ &\qquad\leqslant \biggl(\frac{1}{2\pi}\int_{\mathbb{R}}\|R(i\lambda,A)x\|^{2}\, d\lambda\biggr)^{1/2}\biggl(\frac{1}{2\pi}\int_{\mathbb{R}} \|R(i\lambda,A^{*})y\|^{2}\,d\lambda \biggr)^{1/2} \\ &\qquad\leqslant\sqrt{\nu(A)\nu(A^{*})}\,\|x\|\,\|y\|,\qquad x, y \in \mathscr{H},\quad t > 0. \end{aligned} \end{equation*}$$

Now, the vectors $x,y \in \mathscr{H}$ are arbitrary, so we set $y=it G_{A}(t)x$. Then

$$\begin{equation*} \|itG_{A}(t)x\|^{2} \leqslant \sqrt{\nu(A)\nu(A^{*})}\,\|x\|\, \|it G_{A}(t)x\|,\qquad x \in \mathscr{H}. \end{equation*}$$

Consequently, the estimates (1.27), (1.28) hold. The theorem is proved.

Theorem 10 (the estimates (1.29)) immediately follows from Theorem 9.

Theorem 11 (the estimates (1.30)) follows from Theorems 5 and 9.

In connection with Theorem 6, the question of estimating the quantities $\beta_{\min}(A)$, $\beta_{\max}(A)$ for the infinitesimal operator $A$ of an operator semigroup $T :\mathbb{R}_{+} \to \operatorname{End}\mathscr{H}$ plays an important role. These quantities can be infinite, and, moreover, here we construct an example of an operator semigroup whose infinitesimal operator has compact resolvent and linear combinations of whose eigenvectors are dense in the entire space. In this example the numerical range of the infinitesimal operator fills the entire complex plane $\mathbb{C}$.

Example 1.  An operator semigroup $T\colon \mathbb{R}_{+} \to \operatorname{End}\mathscr{H}$, where $\mathscr{H}=L^{2}(\mathbb{R})$, of the form

$$\begin{equation*} (T(t)x)(s)=\begin{cases} 2x(s+t), & s \in [-t,0], \\ x(s+t), & s \in \mathbb{R} \setminus [-t,0], \end{cases} \end{equation*}$$

is such that $\beta_{\max}(A)=\infty$ for its infinitesimal operator $A$. This fact follows from the equations $\|T(t)\|=2$, $t\in \mathbb{R}$, and Theorem 5.

Example 2.  We consider the Hilbert space $\mathscr{H}=\bigoplus_{n \geqslant 2} \mathbb{C}^{n}$ of sequences $x=(x_{2},x_{3},\ldots)$, where $x_{n} \in \mathbb{C}^{n}$, $n\geqslant 2$, with the inner product $\langle x,y\rangle=\sum^{\infty}_{n=2}(x_{n},y_{n})$, $x_{n},y_{n} \in \mathbb{C}^{n}$. We define a sequence $\mathscr{N}_{n} \in \operatorname{End}\mathbb{C}^{n}$, $n\geqslant 2$, of nilpotent operators by the equations

$$\begin{equation*} \mathscr{N}_{n}x=\biggl(\,\sum^{n}_{k=2}x_{k},0,\dots,0\biggr) \in\mathbb{C}^{n},\qquad n \geqslant 2. \end{equation*}$$

The nilpotency index of every operator $\mathscr{N}_{n}$ is equal to two, that is, $\mathscr{N}_{n}^{2}=0$. We consider the family of vectors $x(\alpha)=(x_{1}(\alpha),\dots,x_{n}(\alpha)) \in \mathbb{C}^{n}$, $\alpha \in \mathbb{T}$, where

$$\begin{equation*} x_{1}(\alpha)=\frac{1}{\sqrt{2}}, \qquad x_{k}(\alpha)=\frac{\alpha}{\sqrt{2}}\,\frac{1}{\sqrt{n-1}}, \quad 2 \leqslant k \leqslant n, \qquad \|x(\alpha)\|=1. \end{equation*}$$

The numerical range $\Theta(\mathscr{N}_{n})$ of the nilpotent operator $\mathscr{N}_{n}$ contains the set of complex numbers of the form

$$\begin{equation*} \bigl(\mathscr{N}_{n}x(\alpha),x(\alpha)\bigr)=\biggl\{\,\sum^{n}_{k=2} x_{k}(\alpha) \overline{x_{1}(\alpha)}; \ \alpha \in \mathbb{T}\biggr\}= \biggl\{\frac{\alpha\sqrt{n-1}}{2}\,; \ \alpha \in \mathbb{T}\biggr\}. \end{equation*}$$

Thus,

$$\begin{equation*} \Theta(\mathscr{N}_{n})\supset \biggl\{\lambda \in\mathbb{C}: |\lambda|=\frac{1}{2}\sqrt{n-1}\biggr\}= \mathbb{T}\biggl(\frac{1}{2}\sqrt{n-1}\biggr), \end{equation*}$$

where the right-hand side is the circle of radius $\frac{1}{2}\sqrt{n-1}$ . By Hausdorff's theorem [13] the set $\Theta(\mathscr{N}_{n})$ is convex and therefore contains the disc

$$\begin{equation*} \mathbb{D}\biggl(\frac{1}{2}\sqrt{n-1}\biggr)= \biggl\{\lambda \in \mathbb{C}:|\lambda| \leqslant \frac{1}{2}\sqrt{n-1}\,\biggr\}. \end{equation*}$$

We further consider the operators

$$\begin{equation*} A_{n}=-\frac{\sqrt{n-1}}{4}I_{n}+\mathscr{N}_{n},\qquad n \geqslant 2, \end{equation*}$$

where $I_{n}$ is the identity operator in $\mathbb{C}^{n}$. From these operators we construct an operator semigroup $T\colon \mathbb{R}_{+} \to\operatorname{End}\mathscr{H}$ by the formula

$$\begin{equation*} T(t)x=(e^{tA_{2}}x_{2},e^{t A_{2}}x_{3},\dots),\qquad x=(x_{n})\in \mathscr{H},\quad t \geqslant 0. \end{equation*}$$

Since

$$\begin{equation*} e^{tA_{n}}=\exp\biggl\{-\frac{\sqrt{n-1}}{4}t\biggr\}(I+\mathscr{N}_{n}), \qquad t\geqslant 0, \end{equation*}$$

it follows that

$$\begin{equation*} \|e^{tA_{n}}\| \leqslant \exp\biggl\{-\frac{\sqrt{n-1}}{4}t\biggr\} \biggl(1+\frac{\sqrt{n-1}}{2}t\biggr),\qquad t \geqslant 0,\quad n \geqslant 2. \end{equation*}$$

Since

$$\begin{equation*} \begin{aligned} \|T(t)x\|&=\biggl(\,\sum_{n \geqslant 2}\|e^{tA_{n}} x_{n}\|^{2}\biggr)^{1/2} \leqslant\sup_{n \geqslant 0}\|e^{tA_{n}}\|\biggl(\,\sum_{n \geqslant 0} \|x_{n}\|^{2}\biggr)^{1/2} \\ &\leqslant \max_{a \geqslant 0}\exp\biggl\{\frac{-at}{4}\biggr\}\biggl(1+\frac{at}{2}\biggr) \|x\|=\frac{2}{\sqrt{e}}\|x\|, \qquad t \geqslant 0,\quad x \in \mathscr{H}, \end{aligned} \end{equation*}$$

it follows that the operator semigroup $T$ is bounded, and

$$\begin{equation*} \sup_{n \geqslant 0} \|T(t)\| \leqslant \dfrac{2}{\sqrt{e}}. \end{equation*}$$

It is easy to see that $\|T(t)\|\leqslant Me^{-t/4}$, $t \geqslant 0$, that is, the semigroup constructed above is exponentially stable. In view of the fact that the semigroup $T$ is continuous on every finite-dimensional subspace $\mathscr{H}_{n}=\{x=(x_{k}) \in \mathscr{H}:x_{k}=0, \ k \geqslant n+1\}$, $n\geqslant 2$, and this system of subspaces is dense in $\mathscr{H}$, the boundedness of the operator semigroup $T$ implies that it is strongly continuous on $\mathscr{H}$.

Let $A$ be the infinitesimal operator of the operator semigroup $T\colon \mathbb{R}_{+} \to \operatorname{End}\mathscr{H}$ constructed above. Since the subspaces $\mathscr{H}_{n}$, $n \geqslant 2$, are invariant for $A$, the numerical range $W_{n}$ of the restriction $A\mid\mathscr{H}_{n}$ of the operator $A$ to $\mathscr{H}_{n}$ is contained in the numerical range $\Theta(A)$ of the operator $A$. Since

$$\begin{equation*} W_{n} \supset \bigcup^{n}_{k=2} \Theta(A_{k}) \supset \bigcup^{n}_{k=2}\biggl\{\lambda \in \mathbb{C}: \biggl|\lambda+\frac{\sqrt{k-1}}{4}\biggr| \leqslant\frac{\sqrt{k-1}}{2}\biggr\} \end{equation*}$$

(we have used the equation $\Theta(A_{k})=-{\sqrt{k-1}}/{4}+\Theta(\mathscr{N}_{k})$) and since the set

$$\begin{equation*} \bigcup^{\infty}_{k=2} \biggl\{\lambda \in \mathbb{C}: \biggl|\lambda+\frac{\sqrt{k-1}}{4}\biggr|\leqslant \frac{\sqrt{k-1}}{2}\biggr\} \end{equation*}$$

coincides with $\mathbb{C}$, it follows that $\Theta(A)=\mathbb{C}$. It follows from the definition of the operator $A$ that the eigenvectors and generalized eigenvectors of this operator generate the space $\mathscr{H}$, and the operator $A$ has compact resolvent.

We now consider the sequence of operators $\widetilde{A}_{n}= A_{n}+\Lambda_{n} \in \operatorname{End}\mathbb{C}^{n}$, $n \geqslant 2$, where $\Lambda_{n}x=(x_{k}/k)$, $x=(x_{1},\dots,x_{n}) \in \mathbb{C}^{n}$. We extend the definition of the operator semigroup $\widetilde{T}\colon \mathbb{R}_{+} \to \operatorname{End}\mathscr{H}$ by the equations

$$\begin{equation*} \widetilde{T}(t)x=(e^{t\widetilde{A}_{2}}x_{2}, e^{t\widetilde{A}_{3}}x_{3},\dots) \qquad x=(x_{n}) \in \mathscr{H},\quad t \geqslant 0. \end{equation*}$$

Its infinitesimal operator is the operator $\widetilde{A}=A+ \Lambda\colon D(A) \subset \mathscr{H} \to \mathscr{H}$, where $\Lambda \in \operatorname{End}\mathscr{H}$, $\Lambda x=(\Lambda_{n} x_{n})$, $x=(x_{n}) \in \mathscr{H}$. This operator semigroup is strongly continuous (since $\widetilde{A}$ is a bounded perturbation of the operator $A$), the numerical range $\Theta(\widetilde{A})$ of $\widetilde{A}$ coincides with $\mathbb{C}$, the resolvent of $\widetilde{A}$ is compact, and the linear hull of the eigenvectors of $\widetilde{A}$ is dense in the space $\mathscr{H}$ ($\widetilde{A}$ has no generalized eigenvectors).

Let $A \in LRC(\mathscr{X})$ be a hyperbolic relation on a Banach space $\mathscr{X}$, that is, condition (1.31) holds. Then the spectrum $\sigma(A)$ of $A$ can be represented in the form (1.32). We consider the quantities $r_{int}\geqslant 0$ and $0 \leqslant r_{out} \leqslant\infty$ setting $r_{int}=0$ if $\sigma_{int}=\varnothing$, and $r_{int}=\max\{|\lambda|; \ \lambda \in \sigma_{int}\}=r(A_{0})$ if $\sigma_{int} \ne \varnothing$. We set $r_{out}=\infty$ if $\sigma_{out}=\varnothing$, and $r_{out}=\max\{|\lambda|; \ \lambda \in \sigma_{out}\}$ if $\sigma_{out} \ne \varnothing$.

The resolvent $R(\,\cdot\,,A)\colon \rho(A) \to \operatorname{End}\mathscr{X}$ of the relation $A$ is a pseudoresolvent (see [1]), which by condition (1.31) of Theorem 14 is holomorphic on the open set $\{\lambda \in \mathbb{C}:r_{int} < \rho < r_{out}\}$. Therefore, using the functional calculus for linear relations [25] or Theorem 5.9.3 in [1] for pseudoresolvents (see also the proof of Theorem 6.1 in [27]), we obtain the representation

$$\begin{equation} G(n)=\frac{1}{2\pi i}\int_{\mathbb{T}(\rho)}\gamma^{n} R(\gamma,A)\,d\gamma,\qquad n \in \mathbb{Z}, \end{equation} \tag{ 5.1 }$$

where $\mathbb{T}(\rho)=\{\lambda \in \mathbb{C}:|\lambda|=\rho\}$ is the circle of radius $\rho$ satisfying $r_{int}<\rho< r_{out}$.

For $n=0$ we obtain the formula (see also formula (1.34))

$$\begin{equation*} P_{0}=-\frac{1}{2\pi i}\int_{\mathbb{T}(\rho)}R(\gamma,A)\,d\gamma \end{equation*}$$

for the Riesz projector $P_{0} \in \operatorname{End}\mathscr{X}$ constructed with respect to the spectral set $\sigma_{int}$.

First we suppose that $\sigma_{int} \ne \varnothing$ and estimate the norms $\|G(n)\|$ of the operators $G(n)$, $n \leqslant 0$.

It follows from the representation (1.37) of the Green's function $G$ that

$$\begin{equation*} \|G(n)\| \leqslant \rho^{n+ 1}\max_{\gamma \in \mathbb{T}} \|R(\gamma\rho,A)\|,\qquad n \leqslant 0, \end{equation*}$$

where $\rho \in (r_{int},1)$. We express the relation $A-\gamma\rho I$, where $\gamma \in \mathbb{T}$, $\rho \in (r_{int},1)$, in the following form:

$$\begin{equation} A-\gamma\rho I=(A-\gamma I)\bigl(I-(-\gamma)(1-\rho)R(\gamma,A)\bigr). \end{equation} \tag{ 5.2 }$$

Since the operator

$$\begin{equation*} C=(-\gamma)(1-\rho)R(\gamma, A) \end{equation*}$$

satisfies $\|C\| \leqslant (1-\rho)\gamma_{\mathbb{T}}(A)$, $\gamma \in \mathbb{T}$, it follows that $\|C\|<1$ for $\rho \in (1-1/{\gamma_{\mathbb{T}}(A)},1)$. Since the spectral radius of a bounded operator does not exceed its norm, we have the inequality

$$\begin{equation*} \begin{aligned} \gamma_{\mathbb{T}}(A)&=\sup_{\gamma \in \mathbb{T}}\|R(\gamma,A)\| \geqslant \sup_{\gamma \in \mathbb{T}}r(R(\gamma,A)) \\ &=\sup_{\gamma \in \mathbb{T}, \ \lambda \in \sigma(A)} \frac{1}{|\gamma-\lambda|}= \frac{1}{\operatorname{dist}(\mathbb{T},\sigma(A))} \geqslant \frac{1}{\operatorname{dist}(\mathbb{T\,},\sigma_{int})}= \frac{1}{1-r_{int}}\,. \end{aligned} \end{equation*}$$

Here we have used the equation

$$\begin{equation} \sigma(R(\gamma,A))=\biggl\{\frac{1}{\lambda-\gamma}\,; \ \lambda \in \sigma(A)\biggr\}, \end{equation} \tag{ 5.3 }$$

which follows from Theorem 2.4 in [25]. Consequently, $r_{int} \leqslant 1-{1}/{\gamma_{\mathbb{T}}(A)}$ and $\gamma_{\mathbb{T}}(A)\geqslant 1$.

Then it follows from equation (5.2) that the relation $A-\gamma\rho I$ is continuously invertible, $(A-\gamma\rho I)^{-1}=(I-C)^{-1}R(\gamma,A)$, and the estimate

$$\begin{equation*} \begin{aligned} \|R(\gamma\rho,A)\|&=\|R(\gamma,A)\|\,\|(I-C)^{-1}\| \\ &\leqslant \gamma_{\mathbb{T}}(A)\sum^{\infty}_{k=0} (1-\rho)^{k}\gamma_{\mathbb{T}}(A)^{k}= \frac{\gamma_{\mathbb{T}}(A)}{1-(1-\rho)\gamma_{\mathbb{T}}(A)} \end{aligned} \end{equation*}$$

holds for all $\gamma \in \mathbb{T}$ and $\rho \in(1-{1}/{\gamma_{\mathbb{T}}(A)},1)$. Therefore,

$$\begin{equation} \begin{aligned} \sum_{k \leqslant 0}\|G(k)\| &\leqslant \inf_{\rho \in (1-{1}/{\gamma_{\mathbb{T}}(A)},1)}\,\sum^{\infty}_{k=0} \frac{\rho\gamma_{\mathbb{T}}(A)}{1-(1-\rho)\gamma_{\mathbb{T}}(A)}\rho^{k} \\ &=\inf_{\rho \in (1-{1}/{\gamma_{\mathbb{T}}(A)},1)} \frac{\gamma_{\mathbb{T}}(A)}{(1-\rho)-(1-\rho)^{2}\gamma_{\mathbb{T}}(A)}\,. \end{aligned} \end{equation} \tag{ 5.4 }$$

The infimum is attained at the point $\rho_{\min}=\sqrt{1-{1}/{\gamma_{\mathbb{T}}(A)}}$ . In order to avoid a cumbersome expression for the estimate, we set $\rho=1-{1}/(2\gamma_{\mathbb{T}}(A))$ on the right-hand side of (5.4). Then we have the estimate

$$\begin{equation} \sum_{k \leqslant 0}\|G(k)\| < 4\gamma_{\mathbb{T}}(A)^{2}- 2\gamma_{\mathbb{T}}(A). \end{equation} \tag{ 5.5 }$$

Now suppose that $\sigma_{out} \ne \varnothing$. It follows from (5.3), where $1<|\gamma|<r_{out}$, that

$$\begin{equation*} \begin{aligned} \gamma_{\mathbb{T}}(A)&=\sup_{\gamma \in \mathbb{T}}\|R(\gamma,A)\| \geqslant\sup_{\gamma \in \mathbb{T}}r(R(\gamma, A)) \\ &=\sup_{\gamma \in \mathbb{T},\ \lambda \in \sigma(A)} \frac{1}{|\gamma-\lambda|}\geqslant \frac{1}{\operatorname{dist}(\mathbb{T},\sigma_{out})}= \frac{1}{r_{out}-1}\,. \end{aligned} \end{equation*}$$

Therefore, $r_{out} \geqslant 1+{1}/{\gamma_{\mathbb{T}}(A)}$. We write the relation $A-\gamma\rho I$ in the form

$$\begin{equation*} A-\gamma\rho I=(A-\gamma I)\bigl(I-(\rho-1)\gamma R(\gamma, A)\bigr),\qquad \gamma \in \mathbb{T},\quad \rho \in (1,r_{out}). \end{equation*}$$

Since

$$\begin{equation*} (\rho-1)\|R(\gamma,A)\| \leqslant (\rho-1)\gamma_{\mathbb{T}}(A) <(r_{out}-1)\gamma_{\mathbb{T}}(A) \leqslant 1, \end{equation*}$$

the relation $A-\gamma\rho I$ is continuously invertible for all $\gamma \in \mathbb{T}$, $\rho \in (1,1+{1}/{\gamma_{\mathbb{T}}(A)})$ and

$$\begin{equation*} R(\gamma\rho,A)=\sum_{n=-\infty}^{-1}((\rho-1)\gamma)^{n} R(\gamma,A)^{n+1},\qquad \gamma \in \mathbb{T},\quad \rho \in \biggl(1,1+\frac{1}{\gamma_{\mathbb{T}}(A)}\biggr). \end{equation*}$$

Consequently,

$$\begin{equation*} \|R(\gamma\rho,A)\| \leqslant \sum^{-1}_{n=-\infty}(\rho- 1)^{n} \gamma_{\mathbb{T}}(A)^{n+1}= \frac{\gamma_{\mathbb{T}}(A)}{1-(\rho-1)\gamma_{\mathbb{T}}(A)} \end{equation*}$$

for all $\gamma \in \mathbb{T}$ and $\rho \in (1,1+{1}/{\gamma_{\mathbb{T}}(A)})$. Therefore we obtain the estimate

$$\begin{equation*} \begin{aligned} \sum_{k=-\infty}^{-1}\|G(k)\| &\leqslant \inf_{\rho \in (1,1+ {1}/{\gamma_{\mathbb{T}}(A)})}\, \sum_{k=0}^{\infty}\frac{\rho^{n+1}\gamma_{\mathbb{T}}(A)} {1-(\rho-1)\gamma_{\mathbb{T}}(A)} \\ &=\inf_{\rho \in (1,1+{1}/{\gamma_{\mathbb{T}}(A)})} \frac{\rho\gamma_{\mathbb{T}}(A)} {(\rho-1)(1-(\rho-1)\gamma_{\mathbb{T}}(A))} \end{aligned} \end{equation*}$$

from formula (5.1). We set $\rho=1+{1}/(2\gamma_{\mathbb{T}}(A))$; then

$$\begin{equation} \sum_{k \geqslant 1}\|G(k)\| < 4\gamma_{\mathbb{T}}(A)^{2}+ 2\gamma_{\mathbb{T}}(A). \end{equation} \tag{ 5.6 }$$

The estimates (5.5) and (5.6) obtained above yield the estimates in the theorem that are to be proved. Theorem 14 is proved.

Proof  of Theorem 15 Since the relation $\mathscr{A} \in LRC(\ell^{\infty})$ is continuously invertible, every solution $x_{0}\in \ell^{\infty}(\mathbb{Z},\mathscr{X})$ of the nonlinear difference inclusion (1.39) is a solution of the difference equation

$$\begin{equation} x(n)=\sum_{m=-\infty}^{\infty}G(n-m)f(m,x(m))=(\Phi(x))(n),\qquad n \in \mathbb{Z}. \end{equation} \tag{ 5.7 }$$

The converse is also true. The map $\Phi\colon \ell^{\infty} \to \ell^{\infty}$ defined by the right-hand side of this equation satisfies a Lipschitz condition with constant

$$\begin{equation*} \sum_{k=-\infty}^{\infty}\|G(k)\|\ell < 1. \end{equation*}$$

Consequently, equation (5.7) is solvable by the contraction mapping principle, and its solution is a solution of the difference inclusion (1.39). The theorem is proved.

We consider an operator of the form

$$\begin{equation*} \mathscr{L}=A-B\colon D(\mathscr{L})=D(A) \subset \mathscr{H} \to \mathscr{H}, \end{equation*}$$

where $A\colon D(A) \subset \mathscr{H} \to \mathscr{H}$ is an antiselfadjoint operator (that is, $iA$ is selfadjoint) with domain of definition $D(A)$ and $B$ is an invertible selfadjoint operator in the algebra $\operatorname{End}\mathscr{H}$. By Stone's theorem (see [1], Theorem 22.4.3) the operator $A$ is the infinitesimal operator of some strongly continuous group of isometry operators $U\colon\mathbb{R}\to \operatorname{End}\mathscr{H}$. Consequently, the operator $\mathscr{L}$ is the infinitesimal operator of some operator group $T\colon \mathscr{R}\to \operatorname{End}\mathscr{H}$ of class $C_{0}$ (see [1], Theorem 13.2.1).

If $A$ and $B$ commute, then the operator $A-B$ is a normal operator, and therefore its spectrum $\sigma(A-B)$ is contained in the set

$$\begin{equation*} \bigl\{\lambda-\mu:\lambda \in \sigma(A) \subset i\mathbb{R},\ \mu \in \sigma(B) \subset \mathbb{R}\bigr\}. \end{equation*}$$

Consequently, $\sigma(\mathscr{L})\cap(i\mathbb{R})=\varnothing$ and $\operatorname{dist}(\sigma(\mathscr{L}), \ i\mathbb{R}) \geqslant\|B^{-1}\|=r(B^{-1})=\sup\{\alpha^{-1}:\alpha \in \sigma(B)\}$. Therefore, $\gamma(\mathscr{L})=\sup_{\lambda \in \mathbb{R}} \|R(i\lambda,\mathscr{L})\|\leqslant \|B^{-1}\|$. Thus, by Theorem 2 (conditions (1.4) and (1.5) hold) the operator semigroup $T$ is hyperbolic. If the operators $A$ and $B$ do not commute, then this result, generally speaking, is not true.

We introduce some quantities and concepts in terms of which we shall state the result on hyperbolicity conditions for the operator semigroup $T$ under consideration.

We consider the two sets

$$\begin{equation*} \sigma_{+}=\sigma_{+}(B)=\{\lambda \in \sigma(B):\lambda> 0\},\qquad \sigma_{-}=\sigma_{-}(B)=\{\lambda \in \sigma(B):\lambda <0\} \end{equation*}$$

and the two quantities

$$\begin{equation*} {\unicode{230}}(B)={\unicode{230}}_{+}(B)+{\unicode{230}}_{-}(B),\qquad {\unicode{230}}(B)=\{{\unicode{230}}_{-}(B), \ {\unicode{230}}_{+}(B)\}, \end{equation*}$$

where

$$\begin{equation*} {\unicode{230}}_{+}(B)=\min\{\lambda;\lambda \in \sigma_{+}\}, \qquad {\unicode{230}}_{-}(B)=|\max\{\lambda;\lambda \in \sigma_{-}\}|, \end{equation*}$$

that is, ${\unicode{230}}_{\pm}(B)=\operatorname{dist}(\sigma_{\pm}, \ i\mathbb{R})$ and $\sigma(B)=\sigma_{+} \cup \sigma_{-}$. In what follows we assume that both $\sigma_{+}$ and $\sigma_{-}$ are non-empty sets.

Let $P_{+}=P(\sigma_{+},B)$ and $P_{-}=P(\sigma_{-},B)$ denote the Riesz projectors constructed with respect to the sets $\sigma_{+}$ and $\sigma_{-}$, respectively. Since the operator $B$ is selfadjoint, these projectors are orthogonal.

Along with the operator $A$ we consider the transformer (operator in the space of operators) $\operatorname{ad}_{A}\colon D(\operatorname{ad}_{A}) \subset \operatorname{End}\mathscr{H}\to\operatorname{End}\mathscr{H}$ with domain of definition $D(\operatorname{ad}_{A})$ consisting of operators $X \in \operatorname{End}\mathscr{H}$ that take the domain of definition $D(A)$ into the domain of definition $D(A)$; furthermore, the operator $AX-XA$ admits an extension with domain of definition $D(A)$ to some operator $Y$ in $\operatorname{End}\mathscr{H}$, which we denote by the same symbol $AX-XA$ or $[A,X]$. Furthermore, we set $\operatorname{ad}_{A}X=Y$.

Theorem 19. (see [32]) Suppose that an invertible operator $B$ belongs to the domain of definition $D(\operatorname{ad}_{A})$ of the transformer $\operatorname{ad}_{A}$ and one of the following conditions holds:

• 1)  
  $\|AB-BA\| < {\unicode{230}}_{0}(B){\unicode{230}}(B)$,
• 2)  
  $\|[B, AB-BA]\| < {\unicode{230}}_{0}(B){\unicode{230}}(B)^{2}$,
• 3)  
  $\|ad^{n}_{B}(AB-BA)\| < {\unicode{230}}_{0}(B){\unicode{230}}(B)^{n+1}$, $n\geqslant 2$,

where $ad^{n}_{B}$ is the $n$th power of the bounded transformer $\operatorname{ad}_{B}\colon\! \operatorname{End}\mathscr{H}\,{\dashrightarrow}\, \operatorname{End}\mathscr{H}$ defined by the formula $\operatorname{ad}_{B}X=BX-XB$, $X \in \operatorname{End}\mathscr{H}$. Then the semigroup $T$ (whose infinitesimal operator is the operator $\mathscr{L=A-B}$) is hyperbolic and the following estimates hold (corresponding to each of the conditions of the theorem):

• 1')  
  $\gamma(\mathscr{L}) \leqslant \dfrac{{\unicode{230}}(B)}{{\unicode{230}}_{0}(B){\unicode{230}}(B)-\|AB-BA\|}$ ,
• 2')  
  $\gamma(\mathscr{L}) \leqslant \dfrac{{\unicode{230}}(B)^{2}}{{\unicode{230}}_{0}(B){\unicode{230}}(B)^{2}-\|[B,AB-BA]\|}$ ,
• 3')  
  $\gamma(\mathscr{L}) \leqslant \dfrac{{\unicode{230}}(B)^{n+1}}{{\unicode{230}}_{0}(B){\unicode{230}}(B)^{n+1}-\|ad^{n}_{B} (AB-BA)\|}$ , $n \geqslant 2$,

where $\gamma(\mathscr{L})=\sup_{\lambda \in \mathbb{R}} \|R(i\lambda,\mathscr{L})\|$.

It is important to note that since $A$ is antiselfadjoint and the operator $B$ is selfadjoint we have $\operatorname{Re}(Ax+Bx,x)=(Bx,x)$, $x \in \mathscr{H}$. Therefore,

$$\begin{equation*} \begin{gathered} \beta_{\min}(A+ B)=\min \{\lambda; \ \lambda \in \sigma(B)\}, \\ \beta_{\max}(A+B)=\max\{\lambda; \ \lambda \in \sigma(B)\}=r(B)=\|B\|. \end{gathered} \end{equation*}$$

We consider a concrete example of a differential operator satisfying the conditions of Theorem 19.

Example 3.  Let $A=\frac{d}{dt}\colon D(A)=W^{1}_{2}(\mathbb{R},H) \subset \mathscr{H}\to \mathscr{H}$ be the differentiation operator acting in the Hilbert space $\mathscr{H}=L_{2}(\mathbb{R},H)$ of square-integrable functions defined on $\mathbb{R}$ with values in a complex Hilbert space $H$. Recall that $\langle x,y\rangle=\displaystyle\int_{\mathbb{R}}(x(t),y(t))\,dt$, $x,y\in\mathscr{H}$, is the inner product in $\mathscr{H}$ if $(\,\cdot\,{,}\,\cdot\,)$ is the inner product in $H$. Its domain of definition is the Sobolev space $W^{1}_{2}(\mathbb{R},H)$ contained in $L_{2}(\mathbb{R},H)$. The operator $A$ is the infinitesimal operator of the group of (isometry) operators of shifts of functions in $L_{2}(\mathbb{R},H)$ and therefore is antiselfadjoint.

If $B\in \operatorname{End}\mathscr{H}$ is the operator of multiplication by a bounded continuous function $Q\colon \mathbb{R} \to \operatorname{End}\mathscr{H}$, that is, $(Bx)(t)=Q(t)x(t)$, $x \in \mathscr{H}$, $t \in \mathbb{R}$, then $B$ belongs to $D(\operatorname{ad}_{A})$ if $Q$ is continuously differentiable and its derivative $\dot{Q}\colon \mathbb{R}\to \operatorname{End}\mathscr{H}$ is bounded on $\mathbb{R}$. In this case, $\operatorname{ad}_{A}B$ is the operator of multiplication by the derivative $\dot{Q}$ of $Q$, that is,

$$\begin{equation*} ((\operatorname{ad}_{A}B)x)(t)=\dot{Q}(t)x(t),\qquad t \in \mathbb{R}. \end{equation*}$$

Theorem 19 immediately implies the following theorem.

Theorem 20.  Suppose that $Q\colon \mathbb{R} \!\to\! \operatorname{End} H$ is a continuously differentiable bounded function with bounded derivative $\dot{Q}$ whose values are selfadjoint invertible operators, and suppose that one of the following conditions holds:

• 1)  
  $\displaystyle\sup_{t \in \mathbb{R}}\|\dot{Q}(t)\|<{\unicode{230}}_{0}(Q){\unicode{230}}(Q)$,
• 2)  
  $\sup\|Q(t)\dot{Q}(t)-\dot{Q}(t)Q(t)\|<{\unicode{230}}_{0}(Q){\unicode{230}}(Q)^{2}$,

where

$$\begin{equation*} \begin{aligned} \unicode{230}_{+}(Q)&=\inf_{t \in \mathbb{R}}\min\{\lambda:\lambda \in \sigma(Q(t)) \cap (\mathbb{R}_{+}\}, \\ \unicode{230}_{-}(Q)&=\inf_{t \in \mathbb{R}}\max\{|\lambda|:\lambda \in \sigma(Q(t)) \cap (\mathbb{R}_{-}\},\quad\text{where}\quad \mathbb{R}_{-}=(-\infty,0]. \end{aligned} \end{equation*}$$

Then the differential operator $\mathscr{L}=\frac{d}{dt}-Q\colon W^{1}_{2}(\mathbb{R},H) \subset \mathscr{H}\to\mathscr{H}$ is invertible and the inverse operator admits the estimate corresponding to conditions 1), 2):

•   
  $\|\mathscr{L}^{-1}\| \leqslant \dfrac{{\unicode{230}}(Q)}{{\unicode{230}}_{0}(Q){\unicode{230}}(Q) -\sup_{t\in\mathbb{R}} \|\dot{Q}(t)\|}$ ,
•   
  $\|\mathscr{L}^{-1}\| \leqslant \dfrac{{\unicode{230}}(Q)^{2}} {{\unicode{230}}_{0}(Q){\unicode{230}}(Q)^{2}-\sup_{t\in\mathbb{R}}\|Q(t)\dot{Q}(t)- \dot{Q}(t)Q(t)\|}$ .

We point out that it follows from [2] that the differential operator $\mathscr{L}=\frac {d}{dt}-Q$ under consideration is the infinitesimal operator of the strongly continuous operator group $T_{u}\colon \mathbb{R} \to \operatorname{End}\mathscr{H}$ defined by the equations

$$\begin{equation*} (T_{u}(t)x)(s)=\mathscr{U}(s, \ s-t)x(s-t), \ s,t \in \mathbb{R},\qquad x \in \mathscr{H}, \end{equation*}$$

where $\mathscr{U}(\tau,s)=U(\tau)U(s)^{-1}$, $s,\tau \in \mathbb{R}$, and $U\colon \mathbb{R}\to \operatorname{End} H$ is the Cauchy operator function, that is, $\dot {U}(t)=Q(t)U(t)$, $t \in \mathbb{R}$, $U(0)=I$ (see [40], Ch. III). Under the hypotheses of Theorem 14 the operator semigroup $T_{u}$ is hyperbolic.

The next theorem follows directly from Theorem 20 and the remark made after the statement of Theorem 11.

Theorem 21.  Suppose that a function $Q$ is bounded, continuously differentiable, $\dot{Q} \in C_{b}(\mathbb{R},\operatorname{End} H)$, and condition 2) of Theorem 20 holds. Then the operator

$$\begin{equation*} \mathscr{L}=\frac{d}{dt}-Q\colon C^{1}_{b}(\mathbb{R},H) \subset C_{b} \to C_{b}=C_{b}(\mathbb{R},H), \end{equation*}$$

where $C^{1}_{b}(\mathbb{R},H)= \{x \in C_{b}(\mathbb{R},H):\dot{x} \in C_{b}(\mathbb{R},H)\}$, is continuously invertible and its norm $\|\mathscr{L}^{-1}\|_{\infty}$ admits the estimate

$$\begin{equation*} \|\mathscr{L}^{-1}\|_{\infty} \leqslant 4\bigl(\varphi(Q)+\|Q\|_{\infty}\varphi(Q)^{2}\bigr), \end{equation*}$$

where $\varphi(Q)={\unicode{230}}(Q)^{2}\bigl({\unicode{230}}_{0}(Q){\unicode{230}}(Q)^{2}- \|[Q,\dot{Q}]\|_{\infty}\bigr)$ and $\|[Q,\dot{Q}]\|_{\infty}$ is the norm of the function $Q\dot{Q}-\dot{Q}Q=[Q, \dot{Q}]$ in $C_{b}(\mathbb{R},\operatorname{End}H)$.

To estimate the quantities defined by formulae (1.13)–(1.17) and used in Theorems 6, 7, 9–11, 18, 21, we can apply the method of similar operators described in [34]–[37]. Note that this method can be applied to a wider class of operators than those considered here.

Suppose that $A\colon D(A) \subset \mathscr{H} \to\mathscr{H}$ is a selfadjoint operator, semibounded above, with compact resolvent and that its spectrum $\sigma(A)=\{\lambda_{1},\lambda_{2},\dots\}$ has the properties

$$\begin{equation*} \begin{gathered} d_{n}=\operatorname{dist}(\{\lambda_{n}\},\sigma(A) \setminus \{\lambda_{n}\})\to \infty,\qquad n \to \infty, \\ \cdots \leqslant \lambda_{n} < \lambda_{n-1} < \cdots < \lambda_{1}. \end{gathered} \end{equation*}$$

As $A$ is semibounded above, the quantity $\beta(A)=\max\{\lambda_{n}, \ n \geqslant 1\}=\lambda_{1}$ is finite. Let $P_{n}$, $n \geqslant 1$, be the Riesz projectors (these are orthoprojectors) constructed with respect to the one-point sets $\{\lambda_{n}\}$, $n \geqslant 1$. Then $AP_{n}=\lambda_{n} P_{n}$, $n \geqslant 1$, and $x=\sum_{n \geqslant 1} P_{n}x$, $x \in \mathscr{H}$. The operator $A$ is sectorial, and, moreover, it is the infinitesimal operator of the operator semigroup

$$\begin{equation*} T_{A}(t)x=\sum_{n \geqslant 1} e^{\lambda_{n}t} P_{n}x,\qquad t \geqslant 0,\quad x \in \mathscr{H}, \end{equation*}$$

for which $\|T_{A}(t)\|=e^{\lambda_{1}t}$, $t \geqslant 0$. The operator $i A$ is antiselfadjoint and therefore is the infinitesimal operator of the isometry group

$$\begin{equation*} T_{iA}\colon \mathbb{R} \to \operatorname{End}\mathscr{H},\qquad T_{iA}(t)x=\sum_{n \in \mathbb{Z}} e^{i\lambda_{n}t} P_{n}x,\qquad x \in \mathscr{H},\quad t \in \mathbb{R}. \end{equation*}$$

For any positive integer $m$, we define two transformers in the algebra $\mathscr{H}$

$$\begin{equation*} J_{m}\colon \operatorname{End}\mathscr{H} \to \operatorname{End}\mathscr{H}, \qquad \Gamma_{m}\colon\operatorname{End}\mathscr{H} \to \operatorname{End}\mathscr{H}. \end{equation*}$$

The transformer $J_{m}$ (the operator of block diagonalization) is defined by the formula

$$\begin{equation*} J_{m}X=P_{(m)}XP_{(m)}+\sum_{n \geqslant m+1}P_{n}XP_{n},\qquad X \in \operatorname{End}\mathscr{H}, \end{equation*}$$

where the convergence of the series is understood in the strong operator topology and the projector $P_{(m)}$ has the form $P_{(m)}=\sum^{m}_{k=1} P_{k}$. It is easy to show that $J_{m}$ is a projector and $\|J_{m}\|=1$ (see [36], [37]); furthermore, the series in the definition of $JX$ is unconditionally strongly convergent. If the image $\operatorname{Im} P_{n}$ of every projector $P_{n}$, $n \geqslant 1$, is one-dimensional, then $P_{n}x=(x, e_{n})e_{n}$, $n \geqslant 1$, where $e_{n}$ is an eigenvector of the operator $A$ corresponding to the eigenvalue $\lambda_{n}$. In this case, the matrix of the operator $Y=J_{m}X$ with respect to the orthonormal basis $\{e_{n}, \ n \geqslant 1\}$ has the form $(y_{ij=1})_{i,j}^{\infty}$, where $y_{ij}=\delta_{ij}x_{ij}$ and $\delta_{ij}$ is the Kronecker delta for $\max\{i,j\} \geqslant m+1$, and $y_{ij}=x_{ij}$ if $\max\{i,j\} \leqslant m$. Here, $x_{ij}=(X e_{j},e_{i})$, $i,j \geqslant 1$, is the matrix of an operator $X \in \operatorname{End}\mathscr{H}$.

We define the transformer $\Gamma_{m}$ using the transformer $\operatorname{ad}_{A}\colon D(\operatorname{ad}_{A}) \subset \operatorname{End}\mathscr{H} \to \operatorname{End}\mathscr{H}$,

$$\begin{equation*} \operatorname{ad}_{A}X=AX-XA,\qquad X \in D(\operatorname{ad}_{A}), \end{equation*}$$

with domain of definition $D(\operatorname{ad}_{A})$ consisting of operators $X \in \operatorname{End}\mathscr{H}$ that have the following properties:

• 1)  
  $X\, D(A) \subset D(A)$,
• 2)  
  the operator $AX-XA\colon D(A) \to \mathscr{H}$ admits a bounded extension $Y$ to the space $\mathscr{H}$ (and we set $\operatorname{ad}_{A}X=Y)$.

The transformer $\Gamma_{m}$ is defined on every operator $X \in \operatorname{End}\mathscr{H}$ as the (unique) solution $Y_{m}$ of the equation

$$\begin{equation*} AY-Y\!A=X-J_{m}X \end{equation*}$$

satisfying the condition $J_{m}Y_{m}=0$. Thus, $(\Gamma_{m}X) D(A) \subset D(A)$ for any operator $X \in \operatorname{End}\mathscr{H}$. The transformer $\Gamma_{m}$ is a bounded operator, and the estimate

$$\begin{equation*} \|\Gamma_{m}\| \leqslant \frac{\pi}{2}\max\{\alpha^{-1}_{m},\beta^{-1}_{m}\} \end{equation*}$$

holds (see [34], [36]), where $\alpha_{m}=\operatorname{dist}(\sigma_{m},\sigma(A) \setminus \sigma_{m})$ and $\beta_{m}=\min\{\lambda_{i}-\lambda_{j},\ {i \ne j}, i,j \geqslant m+1\}$. Consequently, $\lim_{m \to \infty}\|\Gamma_{m}\|=0$. In addition we point out that the operator $\Gamma_{m} X$, where $X \in \operatorname{End}\mathscr{H}$, is completely determined by operators of the form (operator blocks)

$$\begin{equation*} P_{i}(\Gamma_{m}X)P_{j}=\begin{cases} \dfrac{1}{\lambda_{i}-\lambda_{j}}P_{i} X P_{j}, & \max\{i,j\}\geqslant m+1, \ i \ne j, \\ 0, & \max\{i,j\} \leqslant m, \\ 0, & i=j. \end{cases} \end{equation*}$$

Consequently, if $\lambda_{k}$, $k \geqslant 1$, are simple eigenvalues of the operator $A$, then the matrix $(y_{ij})^{\infty}_{i,j=1}$ of the operator $Y=\Gamma X$ has the form

$$\begin{equation*} y_{ij}=\begin{cases} 0, & i,j \leqslant m, \\ \dfrac{x_{ij}}{\lambda_{i}-\lambda_{j}}\,, & \max\{i,j\} \geqslant m+1, \ i \ne j, \\ 0, & i=j, \end{cases} \end{equation*}$$

where $i, j \geqslant 1$ and $x_{ij}=(X e_{j},e_{i})$, $i,j \geqslant 1$, is the matrix of the operator ${X \!\in\! \operatorname{End}\mathscr{H}}$.

Definition 3.  Two linear operators $A_{i}\colon D(A_{i})\subset \mathscr{H} \to \mathscr{H}$, $i=1,2$, are said to be similar if there exists a continuously invertible operator $U \in \operatorname{End}\mathscr{H}$ such that $UD(A_{2})=D(A_{1})$ and $A_{1}U x=UA_{2}x$, $x \in D(A_{2})$. The operator $U$ is called the operator of transformation of $A_{1}$ into the operator $A_{2}$.

Similar operators have a number of coinciding spectral properties. We have the following lemma, which follows immediately from Definition 3.

Lemma 2.  Suppose that $A_{i}\colon D(A_{i}) \subset \mathscr{H} \to \mathscr{H}$, $i=1,2$, are two similar operators and $U \in \operatorname{End}\mathscr{H}$ is an operator of transformation of $A_{1}$ into $A_{2}$. Then the following hold:

1) $\sigma(A_{1})=\sigma(A_{2})$, $\sigma_{d}(A_{1})=\sigma_{d}(A_{2})$, $\sigma_{c}(A_{1})=\sigma_{c}(A_{2})$, $\sigma_{r}(A_{1})=\sigma_{r}(A_{2})$, where $\sigma(A_{i})$, $\sigma_{d}(A_{i})$, $\sigma_{c}(A_{i})$, $\sigma_{r}(A_{i})$, $i=1,2$, are the spectrum, the discrete, continuous, and residual spectra of the operators $A_{i}$, $i=1,2$, respectively;

2) if the operator $A_{2}$ admits a decomposition $A_{2}\!=\!A_{21}\oplus A_{22}$, where ${A_{2k}\!=\!A_{2}\!\mid\! \mathscr{H}_{k}}$, $k=1,2$, is the restriction of $A_{2}$ to $\mathscr{H}_{k}$ with respect to the direct sum ${\mathscr{H}=\mathscr{H}_{1} \oplus \mathscr{H}_{2}}$ of subspaces $\mathscr{H}_{1}$, $\mathscr{H}_{2}$ that are invariant under $A_{2}$, then the subspaces $\widetilde{\mathscr{H}}_{k}=U(\mathscr{H}_{k})$, $k=1,2$, are invariant under the operator $A_{1}$ and $A_{1}=A_{11} \oplus A_{12}$, where ${A_{1k}=A \mid \widetilde{\mathscr{H}}_{k}}$, $k=1,2$, with respect to the decomposition $\mathscr{H}=\widetilde{\mathscr{H}}_{1} \oplus \widetilde{\mathscr{H}}_{2}$; furthermore, if $P_{1}$ is the projector effecting the decomposition $\mathscr{H}=\mathscr{H}_{1} \oplus \mathscr{H}_{2}$ (that is, $\mathscr{H}_{1}=\operatorname{Im}P_{1}$ is the image of the projector $P_{1}$ and $\mathscr{H}_{2}=\operatorname{Im}P_{2}$, where $P_{2}=I-P_{1}$ is the complementary projector of $P_{1}$), then the projector $\widetilde{P}_{1}$ effecting the decomposition $\mathscr{H}=\widetilde{\mathscr{H}}_{1}\oplus\widetilde{\mathscr{H}}_{2}$, is defined by the formula

$$\begin{equation*} \widetilde{P}_{1}=U P_{1} U^{-1}; \end{equation*}$$

3) if $A_{2}$ is the infinitesimal operator of a strongly continuous operator semigroup $T_{2}\colon \mathbb{R}_{+} \to \operatorname{End}\mathscr{H}_{2}$, then the operator $A_{1}$ is the infinitesimal operator of the strongly continuous operator semigroup

$$\begin{equation*} T_{1}\colon \mathbb{R}_{+} \to \operatorname{End}\mathscr{H}_{1},\qquad T_{1}(t)=UT_{2}(t)U^{-1},\quad t \geqslant 0. \end{equation*}$$

The following result (see [34]–[37]) is the main result where the method of similar operators is used in problems concerning obtaining estimates for the parameters of exponential dichotomy and the Green's function of a perturbed selfadjoint operator $A$ with the properties listed above.

Theorem 22.  Suppose that $B \in \operatorname{End}\mathscr{H}$ and a positive integer $m$ is chosen in such a way that the condition

$$\begin{equation} 4\gamma_{m}\|B\|=2\pi\max\{\alpha^{-1}_{m},\beta^{-1}_{m}\}\|B\| < 1 \end{equation} \tag{ 6.1 }$$

holds. Then the operator $A-B$ is similar to the operator $A-J_{m} X_{*}$, where the operator $X_{*} \in \operatorname{End}\mathscr{H}$ is a solution of the (nonlinear) equation

$$\begin{equation} X=B\Gamma_{m}X-(\Gamma_{m}X)(J_{m}B)-(\Gamma_{m}X)J_{m}(B\Gamma_{m}X)+B =\Phi_{m}(X). \end{equation} \tag{ 6.2 }$$

This solution can be found using the method of simple iterations by setting $X_{0}=0$, $X_{1}=B,\ldots$ (the operator $\Phi_{m}\colon \operatorname{End}\mathscr{H} \to \operatorname{End}\mathscr{H}$ is a contraction in the ball $\{X \in \operatorname{End}\mathscr{H}:\|X-B\| \leqslant 3\|B\|\}$, and the operator $X_{*}$ is contained in this ball). A similarity transformation of the operator $A-B$ into the operator $A-J_{m}X_{*}$ is effected by the operator $I+\Gamma_{m}X_{*} \in \operatorname{End}\mathscr{H}$, that is,

$$\begin{equation*} (A-B)(I+ \Gamma_{m}X_{*})=(I+\Gamma_{m}X_{*})(A-J_{m}X_{*}), \end{equation*}$$

Furthermore, $\|\Gamma_{m}X_{*}\| < 1$.

Theorem 18 and Lemma 2 imply the following.

Theorem 23.  Suppose that $B \in \operatorname{End}\mathscr{H}$ and a positive integer $m$ satisfies (6.1). The operator $A-B$ and the similar operator $A(m)=A-J_{m}X_{*}$ (by Theorem 18) are the infinitesimal operators of the analytic operator semigroups $T_{A-B}$, ${T_{A(m)}\colon \mathbb{R}_{+} \to \operatorname{End}\mathscr{H}}$, respectively, and the equations

$$\begin{equation} T_{A-B}(t)=(I+\Gamma_{m}X_{*})^{-1}T_{A(m)}(t)(I+\Gamma_{m}X_{*}),\qquad t \geqslant 0, \end{equation} \tag{ 6.3 }$$

hold, where $X_{*}$ is a solution of the nonlinear equation (6.2) satisfying the condition $\|X_{*}-B\| \leqslant 3\|B\|$. If the operator semigroup $T_{A(m)}$ is hyperbolic, then the operator semigroup $T_{A-B}$ is also hyperbolic and the Green's functions $G_{A(m)}$, ${G_{A-B}\colon \mathbb{R} \to \operatorname{End}\mathscr{H}}$ constructed for them are connected by the similarity relation

$$\begin{equation} G_{A-B}(t)=(I+\Gamma_{m}X_{*})^{-1}G_{A(m)}(t)(I+\Gamma_{m}X_{*}),\qquad t\in \mathbb{R}, \end{equation} \tag{ 6.4 }$$

and

$$\begin{equation} \|G_{A-B}(t) \| \leqslant \frac{1+4\gamma_{m}\|B\|}{1-4\gamma_{m}\|B\|} \|G_{A(m)}(t)\|,\qquad t \in\mathbb{R}. \end{equation} \tag{ 6.5 }$$

Thus, because of (6.3) and (6.4), the method of similar operators makes it possible to reduce studying the operator semigroup $T_{A-B}$ to studying the operator semigroup $T_{A(m)}$ and to obtain estimates for the Green's function $G_{A-B}$ by using estimates of the Green's function $G_{A(m)}$ under the condition that the operator semigroup $T_{A(m)}$ is hyperbolic.

It follows from the form of the operator $A-J_{m}X_{*}$ that it commutes with all the projectors $P_{(m)}=\sum^{m}_{k=1}P_{k},P_{j}$, $j\geqslant m+1$. Consequently, the subspaces ${\mathscr{H}_{(m)}=\operatorname{Im}P_{(m)}}$, $\mathscr{H}_{j}=\operatorname{Im}P_{j}$, $j \geqslant m+1$, are invariant under the operator ${A-J_{m}X_{*}}$. These subspaces also have the same invariance property for the operators $T_{A(m)}(t)$, $t \geqslant 0$.

The Hilbert space $\mathscr{H}$ can be represented in the form

$$\begin{equation*} \mathscr{H}=\mathscr{H}_{(m)} \oplus \mathscr{H}^{(m)},\qquad \mathscr{H}^{(m)}=\bigoplus^{\infty}_{k=m+1} \mathscr{H}_{k}, \end{equation*}$$

which is an orthogonal direct sum of the subspaces $\mathscr{H}_{(m)}$, $\mathscr{H}_{k}$, $k \geqslant m+1$. Therefore the operator semigroup $T_{A(m)}$ and the Green's function $G_{A(m)}$ (under the condition that $T_{A(m)}$ is hyperbolic) admit decompositions of the form

$$\begin{equation} T_{A(m)}(t)=e^{A_{(m)}t}\oplus e^{A_{m+1}t}\oplus e^{A_{m+2}t}\oplus\cdots, \end{equation} \tag{ 6.6 }$$

$$\begin{equation} G_{A(m)}(t)=G_{(m)}(t)\oplus G_{A_{m+1}}(t)\oplus G_{A_{m+2}}(t)\oplus\cdots, \end{equation} \tag{ 6.7 }$$

where $A_{(m)}=A(m)\mid\mathscr{H}_{(m)}$, $A_{m+1}=A(m)\mid\mathscr{H}_{m+1}$, $A_{m+2}=A(m)\mid \mathscr{H}_{m+2}, \dots$, the Green's function $G_{(m)}$ is constructed for the operator $A_{(m)}$, $G_{A_{m+1}}$ is constructed for the operator $A_{m+1}$, and so on. Note that the operators $A_{(m)},A_{m+1},\dots$ act in the finite-dimensional spaces $\mathscr{H}_{(m)},\mathscr{H}_{m+1},\dots$, respectively; furthermore, it follows from Theorem 18 (from equation (6.2)) that these operators are determined by the equations

$$\begin{equation} \begin{gathered} A_{(m)}=P_{(m)} AP_{(m)}-P_{m}X_{*}P_{m}\colon \mathscr{H}_{(m)} \to \mathscr{H}_{(m)}, \\ A_{k}=P_{k} AP_{k}-P_{k} X_{*} P_{k}=\lambda_{k}P_{k}-P_{k}BP_{k}- P_{k}(X_{*}-B) P_{k} \in \operatorname{End}\mathscr{H}_{k}, \\ k \geqslant m+1. \end{gathered} \end{equation} \tag{ 6.8 }$$

Equation (6.2), whose solution is the operator $X_{*}$, and the definition of the transformers $\Gamma_{m}$, $J_{m}$ imply the equations

$$\begin{equation*} P_{k}(X_{*}-B)P_{k}=P_{k}B\Gamma_{m}(X_{*}P_{k}),\qquad k\geqslant m+1. \end{equation*}$$

Since (we use the equations $\Gamma_{m}(P_{j}X_{*}P_{k})=(\lambda_{j}-\lambda_{k})^{-1}P_{j}X_{*}P_{k}$, $j \geqslant m+1$, $j \ne k$)

$$\begin{equation*} \begin{aligned} &\|\Gamma_{m}(X_{*}P_{k})x \|^{2}=\biggl\|\,\sum_{j\geqslant m+1, \ j \ne k} \frac{1}{\lambda_{j}-\lambda_{k}}P_{j} X_{*} P_{k} x\biggr\|^{2} \\ &\qquad=\sum_{j\geqslant m+1, \ j \ne k} \frac{1}{(\lambda_{j}-\lambda_{k})^{2}}\|X_{*} P_{k} x\|^{2} \leqslant \biggl(\,\sum_{j\geqslant m+1, \ j \ne k} \frac{1}{(\lambda_{j}-\lambda_{k})^{2}}\|P_{k} x\|^{2}\biggr)\|X\|_{*} \\ &\qquad\leqslant \frac{1}{\inf_{j \geqslant m+1, \ j \ne k} |\lambda_{k}-\lambda_{j}|^{2}}\|X_{*}\|^{2}\|x\|^{2}= \frac{1}{d_{k, m}}\|X_{*}\|^{2}\|x\|^{2},\qquad x \in \mathscr{H}, \end{aligned} \end{equation*}$$

it follows that $\|\Gamma_{m}(X_{*}P_{k})\| \leqslant d^{-1}_{k}\|X_{*}\|$, $k \geqslant m+1$, and therefore,

$$\begin{equation} \|P_{k}(X_{*}-B) P_{k}\| \leqslant d^{-1}_{k,m}\|B\|\,\|X_{*}\| \leqslant 4d^{-1}_{k,m}\|B\|^{2},\qquad k \geqslant m+1. \end{equation} \tag{ 6.9 }$$

It follows from the representation (6.7) of the Green's function $G_{A(m)}$ of the operator $A(m)$ that

$$\begin{equation*} \begin{aligned} \|G_{A(m)}(t)x\|^{2}&=\|G_{(m)}(t)P_{(m)}x\|^{2}+\sum_{k \geqslant m+1} \|G_{k}(t) P_{k}x\|^{2} \\ &\leqslant \Bigl(\,\sup_{k \geqslant m+1}\bigl\{\|G_{(m)}(t)\|^{2}, \|G_{k}(t)\|^{2}\bigr\}\Bigr)\|x\|^{2},\qquad x \in \mathscr{H}. \end{aligned} \end{equation*}$$

Therefore we have the estimates (see (6.5))

$$\begin{equation} \begin{aligned} \|G_{A-B}(t)\| &\leqslant \frac{1+4\gamma_{m}\|B\|}{1-4\gamma_{m}\|B\|} \|G_{A(m)}(t)\| \leqslant \|G_{A-B}(t)\| \\ &\leqslant \frac{1+ 4\gamma_{m}\|B\|}{1-4\gamma_{m}\|B\|} \sup_{k \geqslant m+1}\bigl\{\|G_{(m)}(t)\|,\|G_{k}(t)\|\bigr\},\qquad t \in \mathbb{R}. \end{aligned} \end{equation} \tag{ 6.10 }$$

Thus, due to the method of similar operators it is possible to `split' the operator $A-B$ and reduce estimating the Green's function $G_{A-B}$ to estimating the Green's functions $G_{(m)}$, $G_{k}$, $k \geqslant m+1$, which are constructed for operators acting in finite-dimensional spaces. It follows from (6.8), which gives a representation of the operators $A_{k}$, $k \geqslant m+1$, that

$$\begin{equation*} e^{A_{k}t}=e^{\lambda_{k}t} e^{B_{k}t},\qquad t \in \mathbb{R},\quad k\geqslant m+1, \end{equation*}$$

where $B_{k}=P_{k}B P_{k}-C_{k} \in \operatorname{End}\mathscr{H}_{k}$, $C_{k}=P_{k}(X_{*}-B) P_{k}$, $k \geqslant m+1$, and the estimate (6.9) implies that $\|C_{k}\| \leqslant 4d^{-1}_{k} \|B\|^{2}$. Consequently $\| B_{k}\| \leqslant {\|B\|+ 4 d^{-1}_{k}\|B\|^{2}}$. Therefore there exists a positive integer $\ell=\ell(m) \geqslant m+1$ such that

$$\begin{equation} \mid\lambda_{k}\mid > \|B\|+4 d^{-1}_{k}\|B\|^{2},\qquad \lambda_{k} < 0,\quad k \geqslant\ell. \end{equation} \tag{ 6.11 }$$

It follows from these estimates and (6.6) that the operator exponential ${t \mapsto e^{A_{k}t}\colon \mathbb{R}_{+} \to \operatorname{End}\mathscr{H}_{k}}$ decays exponentially and, consequently, $G_{k}(t)=e^{A_{k}t}$; $G_{k}(t)=0$, $t <0$, $k \geqslant \ell$. Therefore the question of whether the operator semigroup $T_{A-B}$ is hyperbolic has been reduced to the question of whether the operator group $e^{A_{(m)}t}$, $e^{A_{k}t}$, $m+1 \leqslant k < \ell$, is hyperbolic. Thus, we have established the following.

Theorem 24.  The operator semigroup $T_{A-B}\colon \mathbb{R}_{+} \to \operatorname{End}\mathscr{H}$ is hyperbolic if and only if the following conditions hold:

$$\begin{equation*} \sigma(A(m)) \cap (i\mathbb{R})=\varnothing,\qquad \sigma(A_{k}) \cap (i\mathbb{R})=\varnothing,\qquad m+1 \leqslant k < \ell, \end{equation*}$$

where the number $\ell$ is such that (6.11) holds. When these conditions hold, the Green's function $G_{A-B}$ admits the estimate (6.10).

We point out that if the number $m$ is chosen to be sufficiently big (satisfying the estimate (6.11), where $\ell=m$, along with condition (6.1)), then the condition that the semigroup $T_{A-B}$ be hyperbolic is equivalent to the validity of the condition $\sigma(A(m)) \cap (i\mathbb{R})=\varnothing$. It is also important to note that the operator $A(m)$ can be represented in the form

$$\begin{equation*} A(m)=A^{0}(m)-X_{*}(m) \in \operatorname{End}\mathscr{H}_{(m)}, \end{equation*}$$

where

$$\begin{equation*} A^{0}(m)=P_{(m)}AP_{(m)}-P_{(m)}B P_{(m)}\colon \mathscr{H}_{(m)}\to\mathscr{H}_{(m)}, \quad X_{*}(m)=P_{(m)}(X_{*}-B)P_{(m)}. \end{equation*}$$

Furthermore, $\lim_{m \to \infty}\|X_{*}(m)\|=0$.

Let $\mathscr{H}=L^{2}[0,1]$ be the Hilbert space of square-integrable complex-valued functions defined on the interval $[0,1]$ with the inner product $(x,y)=\displaystyle\int_{0}^{1}x(t) \overline{y(t)}\,dt$, $x,y \in \mathscr{H}$. An example of an operator satisfying the conditions listed above is given by the differential operator

$$\begin{equation*} A-B=\frac{d^{2}}{dt^{2}}-b\colon D(A-B)= D(A) \subset \mathscr{H} \to \mathscr{H}, \end{equation*}$$

where $A=\frac{d^{2}}{dt^{2}}$, $B \in \operatorname{End}\mathscr{H}$ is the operator of multiplication by an essentially bounded function $b\colon [0,1] \to \mathbb{C}$ (in what follows, $\|b\|_{\infty}=\operatorname{ess}\,\sup_{t \in [0,1]}|b(t)|$), and $D(A)=\{x \in \mathscr{H}: x \ \text{is absolutely continuous and} \ \dot{x} \in L^{2} [0,1]\}$.

The operator $A$ is a selfadjoint operator with compact resolvent. Its spectrum coincides with the set $\{-(\pi n)^{2}, \ n \geqslant 1\}$, the eigenfunctions $e_{n}(t)=\sqrt{2}\,\sin \pi nt$, $t\in [0,1]$, $n \geqslant 1$, form an orthonormal basis in $\mathscr{H}=L^{2}[0,1]$, and $A e_{n}=-(\pi n)^{2} e_{n}$, $n \geqslant 1$. The operator $B$ is bounded and $\|B\|=\|b\|_{\infty}$. The Riesz projector $P_{n}$ constructed with respect to the one-point set $\{-\pi^{2}n^{2}\}$ has the form $P_{n}x=(x,e_{n})e_{n}$, $n \geqslant 1$, and $AP_{n}=-(\pi n)^{2}P_{n}$. In this case the quantities used in Theorem 20 have the form

$$\begin{equation*} d_{k}=\pi^{2}(2k -1),\qquad k \geqslant 2,\quad d_{1}=3\pi^{2};\qquad d_{k,m} \geqslant 2m+3. \end{equation*}$$

To estimate the parameters of exponential stability of a strongly continuous operator semigroup $T\colon \mathbb{R}_{+} \to \operatorname{End}\mathscr{H}$, one can use the Hille-Phillips-Yosida theorem (see [1]). The following holds.

Theorem 25.  For numbers $(M,w)$, where $M \geqslant 1$, $w < 0$, to be the parameters of exponential stability of a semigroup $T$ with infinitesimal operator $A$ it is necessary and sufficient that for all $\lambda \in \mathbb{C}_{w}=\{\mu \in \mathbb{C}: \operatorname{Re}\mu >w\}$ and $n \in \mathbb{N}$ the following estimates hold:

$$\begin{equation} \|R(\lambda,A)^{n}\| \leqslant M(\operatorname{Re}\lambda-w)^{-n},\qquad n \geqslant 1,\quad \operatorname{Re}\lambda > w. \end{equation} \tag{ 7.1 }$$

Condition (7.1) is difficult to verify because of the need to estimate the norms of powers of the operators $R(\lambda,A)$, $\lambda \in \mathbb{C}_{w}$.

If $\mathscr{H}$ is a finite-dimensional space and an operator $A \in \operatorname{End}\mathscr{H}$ has spectrum $\sigma(A)=\{\lambda_{1},\dots,\lambda_{m}\}$ with $\operatorname{Re} \lambda_{i} < 0$, $1 \leqslant i \leqslant m$, then the exponential stability of the operator-valued function $t \mapsto e^{At}\colon \mathbb{R}_{+} \to \operatorname{End}\mathscr{H}$ follows from the following Gel'fand-Shilov estimate (see [41], Ch. II, §6, and [40], Ch. I, Exercise 16):

$$\begin{equation} \|e^{At}\| \leqslant \biggl(\,\sum^{n-1}_{k=0} \frac{1}{k!}(2t\|A\|)^{k}\biggr) e^{{\unicode{230}} t}, \qquad t \in \mathbb{R}, \end{equation} \tag{ 7.2 }$$

where ${\unicode{230}}=\max_{1 \leqslant i \leqslant m} \operatorname{Re}\lambda_{i}$ and $n=\dim\mathscr{H}$. This estimate can be used to estimate the parameters of exponential stability of the operator semigroup $T_{A-B}$ in §6. Namely, after a similarity transformation, estimating the Green's function $G_{A-B}$ reduces to estimating the Green's function $G_{A(m)}$ constructed for the operator $A(m)$. Using the estimates (7.2), an estimate for the Green's function under the condition $\sigma(A(m))\subset\mathbb{C}_{-}=\{\lambda \in \mathbb{C}:\operatorname{Re}\lambda < 0\}$ makes it possible to obtain a sufficient condition for the exponential decay of the operator semigroup $T_{A-B}$ and the corresponding estimates on the parameters of exponential stability.

The exponential decay of an operator semigroup $T\colon \mathbb{R}_{+}\to \operatorname{End}\mathscr{H}$ satisfying the condition $\displaystyle\int^{\infty}_{0}\|T(t)x\|^{2}\,dt<\infty$, $x \in \mathscr{H}$, was established by Pazy (see [42], [43]). The principle of uniform boundedness for operators (see [1]) implies that

$$\begin{equation} \nu(T)=\sup_{\|x\| \leqslant 1}\int^{\infty}_{0}\|T(t)x\|^{2}\,dt \end{equation} \tag{ 7.3 }$$

is finite. Since the semigroup $T$ is decaying exponentially, it follows that $G_{A}(t)=T(t)$, $t \geqslant 0$, and therefore it follows from equation (7.3) that $\nu(T)=\nu(A)$. Consequently, we can apply Theorem 7 to estimate this type of operator semigroup. Thus, the following holds.

Theorem 26.  Suppose that an operator semigroup $T\colon \mathbb{R}_{+} \to \operatorname{End}\mathscr{H}$ satisfies the Pazy condition (the quantity (7.3) is finite). Then $T$ is an exponentially stable operator semigroup and the estimate

$$\begin{equation*} \|T(t)\| \leqslant \sqrt{\nu(T)\|W^{-1}\|}\exp\biggl\{-\frac{1}{2\nu(T)}t\biggr\},\qquad t \geqslant0, \end{equation*}$$

holds for this semigroup under the condition that the solution $W \in \operatorname{End}\mathscr{H}$ of Lyapunov's equation (1.23), defined by formula (1.24), is continuously invertible.

Under the condition that the operator $A\colon D(A) \subset \mathscr{H} \to \mathscr{H}$ can be represented in the form $A=A_{0}+ B$, where $A^{*}_{0}=-A_{0}$, $B \in \operatorname{End}\mathscr{H}$, this operator is the infinitesimal operator (see [1]) of a strongly continuous operator group $T\colon \mathbb{R}_{+} \to \operatorname{End}\mathscr{H}$ with numerical range for which $|\beta_{\min}(A)|\leqslant \|B\|$ and $\beta_{\max}(A) \leqslant \|B\|$. Therefore Assumption 1 in §2 holds and, consequently, the assertion of Theorem 6 is true for these operators; also the quantities $|\beta_{\min}(A)|$ and $\beta_{\max}(A)$ (in the estimates (1.21) and (1.22)) can be replaced by the number $\|B\|$.

Commenting on the results of §6 we point out that if a linear relation ${\mathscr{A} \in LRC(\mathscr{X})}$ is a closed linear operator, then it was established in [30] that the operator $\mathscr{A} \in LO(\ell^{p})$ is invertible simultaneously in all the spaces $\ell^{p}=\ell^{p}(\mathbb{Z},\mathscr{X})$, $p \in [1,\infty]$, and the following estimates were obtained for the Green's function $G\colon \mathbb{Z} \to \operatorname{End}\mathscr{X}$:

$$\begin{equation*} \begin{gathered} \|G(k)\| \leqslant \|\mathscr{A}^{-1}\|,\qquad k \in\mathbb{Z}, \\ \|G(k)\|\leqslant 2 \|\mathscr{A}^{-1}\| \biggl(1+ \frac{1}{2\|\mathscr{A}^{-1}\|}\biggr)^{1-k},\qquad k \geqslant 1, \\ \|G(k)\| \leqslant 2 \|\mathscr{A}^{-1}\| \biggl(1-\frac{1}{2\|\mathscr{A}^{-1}\|} \biggr)^{1-k} \end{gathered} \end{equation*}$$

if $\|\mathscr{A}^{-1}\| > 1$, $k \leqslant -1$, and $G(k)=0$ if $\|\mathscr{A}^{-1}\| \leqslant 1$ (if $\mathscr{X}$ is a Hilbert space, then in all these estimates the quantity $\|\mathscr{A}^{-1}\|$ can be replaced by the quantity $\gamma_{\mathbb{T}}(A)$). Furthermore, the estimate

$$\begin{equation*} \|\mathscr{A}^{-1}\|_{q} \leqslant 8\|\mathscr{A}^{-1}\|^{2}_{p}- \|\mathscr{A}^{-1}\|_{p}+1 \end{equation*}$$

is true, where the symbols $\|\mathscr{A}^{-1}\|_{q}$ and $\|\mathscr{A}^{-1}\|_{p}$ denote the norm of $\mathscr{A}^{-1}$ in the spaces $\ell^{q}$ and $\ell^{p}$, respectively. In particular, if $p=2$ and $\mathscr{X}$ is a Hilbert space, then $\|\mathscr{A}^{-1}\|_{2}=\gamma_{\mathbb{T}}(A)$ and therefore,

$$\begin{equation*} \|\mathscr{A}^{-1}\|_{q} \leqslant 8\gamma_{\mathbb{T}}(A)^{2}- \gamma_{\mathbb{T}}(A)+1,\qquad q \in [1,\infty]. \end{equation*}$$

If the operator $A$ belongs to the algebra $\operatorname{End}\mathscr{X}$, then $\mathscr{A} \in \operatorname{End} \ell^{p}$, $p \in [1,\infty]$, and we can use the results in [43]–[47] about estimates for elements of inverse matrices for bounded operators (see also [48]) to estimate the Green's function $G\colon \mathbb{Z}\to \operatorname{End}\mathscr{X}$. For example, Theorem 3 in [43], in particular, gives the estimate $\sum_{k \in \mathbb{Z}}\|G(k)\| \leqslant 16{\unicode{230}}(A)\gamma_{\mathbb{T}}(A)$, where ${\unicode{230}}(A)=\max\{1,\|A\|\}$, under the assumption that $\mathscr{X}$ is a Hilbert space.

For an operator $A \in \operatorname{End}\mathscr{X}$, $\dim\mathscr{X} < \infty$, such that $r(A) < 1$, estimates for the norms of powers of the operator $A$ of the form

$$\begin{equation*} \|A^{k}\| \leqslant \sup_{|\lambda|= r(A)^{1/2}}\|R(\lambda,A)\|(r(A))^{(k+1)/{2}},\qquad k \geqslant 2, \end{equation*}$$

were obtained in the monograph [14], §8.5. Such estimates are also true for ${A \in \operatorname{End}\mathscr{X}}$ in the case of an arbitrary Banach space $\mathscr{X}$ if $r(A)<1$.

Suppose that $\dim\mathscr{X}=m < \infty$ and $A \in \operatorname{End}\mathscr{X}$ with $r(A)<1$. To estimate $\|A^{n}\|$, $n \geqslant m$, we can use the Gel'fand-Shilov method, above, which they applied to obtain the estimate (7.2) for the operator exponential. First suppose that the eigenvalues of the operator $A$ are simple, and let $\sigma(A)=\{\lambda_{1},\dots,\lambda_{m}\}$. Then, in accordance with the Gel'fand-Shilov method based on using the interpolation polynomial in the Newton form, for any $n \geqslant m$ we represent the operator $A^{n}$ in the form

$$\begin{equation*} A^{n}=a_{1}I+a_{2}A+\cdots+a_{m}A^{m-1}, \end{equation*}$$

where the numbers $a_{k} \in \mathbb{C}$, $1 \leqslant k \leqslant m$, admit estimates of the form

$$\begin{equation*} \begin{gathered} |a_{1}| \leqslant \max_{\lambda\in \mathscr{B}}|\lambda^{n}|=r(A)^{n}, \\ |a_{k+1}| \leqslant \frac{1}{k!}n(n-1)\cdots (n-k-1) \max_{\lambda \in \mathscr{B}}|\lambda|^{n-k}=C^{k}_{n} r(A)^{n-k},\qquad 1 \leqslant k \leqslant m-1. \end{gathered} \end{equation*}$$

Here the symbol $\mathscr{B}$ denotes the convex hull in $\mathbb{C}$ of the eigenvalues $\lambda_{1},\dots,\lambda_{m}$ of the operator $A$. If the eigenvalues are not simple, then we have to consider a sequence of operators $(A_{k})$ in $\operatorname{End}\mathscr{X}$ converging to $A$ in which every operator has simple eigenvalues.

The estimates given above imply the following.

Theorem 27.  The following estimates hold

$$\begin{equation*} \|A^{n}\| \leqslant \sum^{m-1}_{k=0}C^{k}_{n} r(A)^{n-k},\qquad n \geqslant m. \end{equation*}$$

Gil' [49] developed a method for estimating the norms of powers of an operator $A \in \operatorname{End} \mathbb{C}^{m}$ (as well as $e^{tA}$) using the quantity

$$\begin{equation*} g(A)=\biggl(N^{2}(A)-\displaystyle\sum^{m}_{k=1} |\lambda_{k}(A)|^{2}\biggr)^{1/2} \leqslant \dfrac{1}{2} N^{2}(A^{*}-A), \end{equation*}$$

where $N(A)=(\operatorname{tr}(AA^{*}))^{1/2}$ is the Hilbert-Schmidt norm of $A$. Namely, the estimates

$$\begin{equation*} \|A^{n}\| \leqslant \sum^{m-1}_{k=0}\frac{\gamma_{m,k}\,n!\, g^{k}(A) r(A)^{n-k}}{(n-k)!\, k!}\,,\qquad n \geqslant 1, \end{equation*}$$

hold, where

$$\begin{equation*} \gamma_{m,k}=\sqrt{\dfrac{C^{k}_{m-1}}{(m-1)!}}, \qquad 1 \leqslant k \leqslant m-1, \quad \gamma_{m,0}=1. \end{equation*}$$

We point out that in Theorems 3–5 in [50] estimates of solutions of differential equations with periodic coefficients were obtained based on using Lyapunov's differential matrix equation. A detailed proof of Theorem 13 is given in [51].