Operator algebra in the space of images

A consistent description of images on the disk and of their transformations is given as elements of a vector space and of an operators algebra. The vector space of images on the disk is the Hilbert space L²() that has as a basis the Zernike functions. To construct the operator algebra that transforms the images, L²() must be complemented and the full rigged Hilbert space RHS() considered. Only this rigged Hilbert space allows indeed to write the operators of different cardinality we need to build the ladder operators on the Zernike functions that by inspection, belong to the representation ${D}_{1/2}^{+}\otimes {D}_{1/2}^{+}$ of the algebra su(1, 1) ⊕ su(1, 1). Consequently the transformations of images are operators contained inside the universal enveloping algebra UEA[su(1, 1) ⊕ su(1, 1)]. Because of limited precision of experimental measures, physical states can be always described by vectors of the Schwartz space (), dense in the L²() space where the manipulation of images is performed.