Spectral analysis of one class perturbed first order differential operators

We use the method of similar operators to study a mixed problem for a first order differential equation with a fractional integration operator. The differential operator defined by the equation is transformed into a similar operator that is an orthogonal direct sum of finite-rank operators. The estimates of eigenvalues, eigenvectors are obtained. The result is used to construct an operator group.

To formulate the problem we introduce the functional spaces. Let H = L 2 = L 2 [0, ω] be the Hilbert space of equivalence classes of square integrable Lebesgue measurable complex functions on closed interval [0, ω]. The inner product on this space is given, as usually, by (x, y) = 1 ω ω 0 x(t)y(t) dt, x, y ∈ H.
The norm in H is induced by this inner product. By W 1 2 [0, ω] we denote the Sobolev space {y ∈ L 2 : y is absolutely continuous and y ∈ L 2 }. By C(J , L 2 ) we shall denote the linear space of all functions v : J × [0, ω] → C such that, for each fixed t ∈ J , the function s → v(t, s) belongs to L 2 and the function v : J → L 2 , ( v(t))(s) = v(t, s), t ∈ J , s ∈ [0, ω], is continuous. If J is a finite interval, then C(J , L 2 ) is a Banach space with the norm v ∞ = max t∈J v(t) 2 . The function v is called associated function to v and they will be identified. The problem (1) in the Hilbert space L 2 in the operator form is written respectively as The operator L : D(L) ⊂ L 2 → L 2 in the equation (2) is defined by The domain D(L) is given by the periodic boundary conditions We note that L = L 0 − B, where L 0 = d/dt, D(L 0 ) = D(L) and the operator B is the fractional integration operator (see [1][2][3][4]) with α > 1/2. The operator L 0 we call the unperturbed or free operator, the operator B we call the perturbation and the operator L we call the perturbed operator. The operator B is called the Riemann-Liouville fractional integration operator and in general y ∈ L 1 [0, ω] (see [1][2][3][4]). In our case (α > 1/2) this operator is a integral operator with square summable kernel on [0, ω] (see [5]).
Recently, interest in the mixed problem for the hyperbolic equation with an involution has intensified (see, for example, [6][7][8][9][10][11][12][13][14][15][16][17][18] and references therein). In these works the problem of justification of Fourier method was studied. There was also studied the asymptotics of eigenvalues and equiconvergence of spectral resolution. In [6][7][8][9][10][11][12] the resolvent method was used to jastily the Fourier method. Another alternative research method for study this problem is the similar operators method. In [13][14][15][16][17][18] the modification of the similar operators method for the first order differential operators with an involution was given. This modification is fully suitable for the study of the problem (1) (without preliminary similary transform), which this paper is devoted to In this paper we study the spectral properties of the operator L, in particular, the problem of bi-invariant subspaces (see Definition 5). Our primary focus is describing the group generated by the operator L.
The similar operators method was pioneered by Friedrichs [19] and then extensively developed and used, for example, in [20][21][22]. This method has many modifications (one can see in [13,20,23,25,26,28]). We note, that this method can be used for the various classes of perturbed linear operators: for the differential operators second order [20,21], for the nonquasianalytic operators [23], for the integro-differential operators [24], for the difference operator with growing potential [25,26]. We use the modification first proposed in [13] then used in [14][15][16][17][18]. This version of the similar operators method is fully formed in [27].

Materials and methods
The main tools for studying the problem under consideration are similar operators and direct sums.
The notation and terminology used herein agree almost completely with that of [14,15,27]. By symbol End H we denote the Banach algebra of all bounded linear operators in H with the norm X ∞ = sup x 1 Xx , x ∈ H, X ∈ End H.
We begin with the definition of the similar operators. Definition 1. Two linear operators A i : D(A i ) ⊂ H → H, i = 1, 2, are called similar if there exists a continuously invertible operator U ∈ End H such that The operator U is called the operator of transformation operator A 1 into A 2 or intertwining operator [2]. The similar operator method is one of the trunsmutation methods. The history, state of the art and of the trunsmutation theory one can be found in [2,[29][30][31].
Similar operators possess a series properties: Moreover, if P is a projection corresponding to the expansion H = H 1 ⊕ H 2 , that is, H 1 = Im P is the image of the projection P , H 2 = Im (I −P ) is the image of the additional projection I −P , then the projection P ∈ End H corresponding to the expansion H = H 1 ⊕ H 2 is defined by 5) if the operator A 2 is a generator of a strongly continuous group of the operators T 2 : R → End H (of class C 0 ), then the operator A 1 is a generator of a strongly continuous group of operators We shall need to extend property 4) to the case of countable direct sum (see also [14,15]). Let the Hilbert space H represented as the direct sum of orthogonal non-zero closed subspaces H n , n ∈ Z, that is where H i is orthogonal to H j as i = j, i, j ∈ Z, and x = n∈Z x n , x n ∈ H n , x 2 = n∈Z x n 2 . In other word, we have a disjunctive resolution of identity that is five properties hold: 1) P * n = P n , n ∈ Z; 2) P i P j = 0 as i = j, i, j ∈ Z; 3) the series n∈Z P n x unconditionally converges to x ∈ H and According to [5, Ch. 5] the system of subspaces H k , k ∈ Z, is an orthogonal basis of subspaces in H.
Definition 3. [14,15] A linear operator A : D(A) ⊂ H → H is called an orthogonal direct sum of bounded operators A n ∈ End H n , n ∈ Z, with respect to resolution (4) and it is written as x k = P x, k ∈ Z} for all n ∈ Z; 2) each subspace H n , n ∈ Z, is invariant with respect to the operator A and A n , n ∈ Z, is the restriction of the operator A on H n , n ∈ Z; We introduce a two-sided ideal of Hilbert-Schmidt operators S 2 (H) in the algebra End H. By X 2 we denote the norm of Hilbert-Schmidt operator X ∈ S 2 (H), that is X 2 = (tr XX * ) 1/2 . Here tr (XX * ) is the trace of the operator XX * belonging to a two-sided ideal S 1 (H) of nuclear operators in End H, with the X 1 = tr (XX * ) = n∈Z s n , where (s n ) is the sequence of s-numbers of an operator X (see [5]).
Definition 4. [14,15] The decomposition of the Hilbert space H, where U is an invertible operator in End H and H is the orthogonal direct sum (4) of subspaces H k , k ∈ Z, will be said to be quasiorthogonal or U -orthogonal. The quasiorthogonal decomposition of the space H is called also a Riesz decomposition.
If the operator U can be represented in the form U = I + W , where W ∈ S 2 (H), then the quasiorthogonal decomposition of the space H is called a Bari decomposition. A linear closed operator A : D(A) ⊂ H → H will be called the quasiorthogonal (U -orthogonal) direct sum of bounded operators A k , k ∈ Z, with respect to the quasiorthogonal decomposition (6) Further, according to [5, Ch. 5], the system of subspaces U H k , k ∈ Z, is a basis of subspaces equivalent to an orthogonal one, or a rectifiable basis [32].
We denote P (k) = |i| k P i , k ∈ Z + . Let I n , n ∈ Z, are the identity operators in the one dimensional space H n = Im P n , n ∈ Z, and I (k) is the identity operator in the subspace The unperturbed operator L 0 is the orthogonal direct sum of the operators (L 0 ) n = L 0 |H n = i2πn ω I n = λ n I n . All the operators (L 0 ) n have rank one. The free operator L 0 is also the orthogonal direct sum of the operators (L 0 ) (k) = L 0 |H (k) , k ∈ Z, and (L 0 ) n = L 0 |H n , |n| > k. The operator (L 0 ) (n) has rank 2k + 1. We have with respect of the representations of the space H as i2πj ω I j with respect to the orthogonal expansion The perturbation B belongs to the ideal S 2 (H), The main result of this paper is that the perturbed operator L is similar to the operator L 0 − B 0 , which is the orthogonal direct sum of the finite rank operators.
Let us apply a similarity transformation from [27] to the original operator (3). The perturbed operator L satisfied for all condition of [27,Theorem 4.5]. We apply this theorem and imply the main result of the present paper. Theorem 1. There exists a number k ∈ Z + , such that the operator L is similar to the operator L 0 − B 0 , where the operator B 0 belongs to the ideal S 2 (H) and the subsets H (k) = Im P (k) , H i = Im P i , k ∈ Z + , |i| > k, are invariant with respect to the operators L 0 , B 0 , L 0 − B 0 . One has L(I + W ) = (I + W )(L 0 − B 0 ) and the operator L is the (I + W )-orthogonal sum. We have The operator I + W is bounded and invertible, W ∈ S 2 (H) and the (I + W )-orthogonal decomposition H = (I + W )H (k) ⊕ (⊕ |j|>k (I + W )H j ) is the Bari decomposition.
Note that the operator B 0 ∈ S 2 (H) has a block-diagonal matrix with respect to the system (5) of spectral projections of the unperturbed operator L 0 . We don't need the preliminary similar transform [27, § 4.2] of the similar operators method. On our case the perturbation B belongs to S 2 (H).
The operator W in Theorem 1 can be effectively calculated as the limit of a sequence of operators obtained by applying the simple iterations method to a Riccati-type equation (see [16, § 3], [14, § 4, 5], [15, §3, 4] for details). The operator B 0 in the Theorem 3 is the solution of the nonlinear operator equation of the similar operators method (see [16, § 3], [14, § 4, 5], [15, §3, 4]). This operator can be founded by the method of simple iterations letting B 0(0) = 0, B 0(1) = B, . . . . Since the similar operators have the same spectrum, we see that Theorem 1 imply the following assertion. Theorem 2. The spectrum σ(L) of the operator L can be represented in the form where the set σ (k) contains at most 2k + 1 eigenvalues, the sets σ j = { λ j }, |j| > k, are singletons and the representation holds with the sequence (β j , |j| > k), such that |j|>k |β j | < ∞. The corresponding eigenvectors e j , j ∈ Z, e j = (I + W )e j , |j| > k, of the operator L form the Bari basis in H and |j|>k e j − e j 2 2 < ∞.
The next theorem is formulated under the assumptions of Theorems 1 and 2 in terms of notations introduced in Theorem 2 (see also [27]). Theorem 3. The limiting relation Let us construct the operator group generated by L (see also [15, § 7], [14, § 6]). Theorem 4. The operator L is the generator a strongly continuous operator group T : R → End H. The group T : R → End H is similar to the group T : R → End H that admits the orthogonal decomposition The group T (t), t ∈ R, admits a (I + W )-orthogonal decomposition with respect to the Bari decomposition H = (I + W )H (k) ⊕ ⊕ |j|>k (I + W )H j of the space H = L 2 .
The analogical theorem for the difference operator with growing potential was prooved in [33].
Let us pass to bi-invariant subspaces. Theorem 6. The subspaces H (k) = (I + W )H (k) , H j = (I + W )H j are the bi-invariant Bari subspaces for the perturbed operator L.
Definition 6. A classical solution to problem (1), where ϕ ∈ W 1 2 is a function u : J ×[0, ω] → C belonging to the space C(J , L 2 ) such that the associated function u : J → L 2 is continuously differentiable and solves problem (1).
Definition 7. A function u : J → L 2 is called a mild solution to problem (1) if there exists a sequence of functions ϕ n ∈ W 1 2 , n 1, such that lim n→∞ ϕ n = ϕ in L 2 and u is a uniform limit on compact subsets J × [0, ω] of a sequence of classical solutions (u n ), n 1, of problem (1) with u n (0, s) = ϕ n (s), s ∈ [0, ω].
Theorem 7. Every classical solution u ∈ C(J , L 2 ) of (1) is given by u(t, s) = ( T (t)ϕ)(s), s ∈ [0, ω], t ∈ J , where ϕ ∈ W 1 2 and ϕ(0) = ϕ(ω). Every mild solution is also given by (7) with ϕ ∈ L 2 . Remark. In the theory of operator simegroups, a mild solution of problem (1) is defined using the group T : R → EndH without using Definition 7 namely, a mild solution is side to be a function of the from u(t, s) = (T (t)ϕ)(s), s ∈ [0, ω], t ∈ R, ϕ ∈ H. Therefore, a theorem that states that this operator is the generator of a strongly continous group of operator T : R → EndH is a theorem on the existence of a mild solution of problem (1).
The existens of the group of operators T makes it possible to correctly define of operator d/dt − L in the Banach space C b (R, L) ⊂ C(R, L) of continuous and R-bouded functions and in ather functional spaces. This enables the use of resalts obtainet in the works [20,[33][34][35][36][37].

Conclusion
We consider the perturbed differencial operator first order L (3) which connected with the mixed problem (1). The main method for this exploration was the method of similar operators, which modification from [27] allowed one to inverstigate the various classes of perturbed differential operators first order. This method allowed us to reduce the operator to one with a block-diagonal matrix. The version of the similar operator mathod from [27] can be applied for many other operators. In this article the asymptotic estimates of eigenvalues, eigenvectors and spectral projections for the operator L (3) were obtained.