Holubchak O. Operators on Hilbert spaces of symmetric analytic functions on a Banach space with a symmetric structure.

In the dissertation work, Hilbert spaces generated by symmetric polynomials and analytic functions on and operator acting on these spaces are investigated. A polynomial on the space of absolutely summing sequences is said to be symmetric if it is invariant with respect to action of the symmetric group on the set of standard basis elements of Combinatorial relations between different algebraic bases of symmetric polynomials in and corresponding generating functions are obtained. Some representations of the set of multisets in spaces of symmetric analytic functions are proposed. Weighted Hilbert spaces of formal series of symmetric polynomials on the space of absolutely summable sequences are constructed. For every the linear multiplicative functional is well-defined on the subspace of all symmetric polynomials in It is continuous, it can be extended to The set of all such that all are continuous form a domain in Every defines a symmetric analytic functions on In the dissertation conditions of the continuity of point evaluation functionals are studied and domains are described. Representations of the Hilbert spaces of symmetric analytic functions in the form of weighted symmetric Fock spaces are considered. For a given algebraic basis of symmetric polynomials on polynomials form a linear basis. If the linear basis is orthonormal in then it can be corresponded to an orthonormal basis in a weighted symmetric Fock spaces. Using these representations, some applications for weighted symmetric Fock spaces are obtained. In particular, using different algebraic bases of symmetric polynomials it is possible to construct nonhomogeneous orthogonal bases in the symmetric Fock space. Operators and functionals which preserve multiplication of polynomial (multiplicative operators and functionals) are investigated. Necessary and sufficient conditions for a given multiplicative functional to be continuous are obtained. For spaces endowed with submultiplicative norm obtained a natural isomorphism to the dual to the symmetric Fock space. There are naturel algebraic operations on quotient set of with respect to the action of the symmetry group. Using these operations one can construct various natural operators of composition. The composition operators on the Hilbert spaces of symmetric analytic functions are investigated. Conditions for the closedness, boundedness, and self-adjointness of such operators are established. A reproducing kernel is constructed in a Hilbert space of symmetric analytic functions on and biorthogonal bases of symmetric polynomials are investigated. Fractional symmetric functions and corresponding composition operators are considered. In the work constructed a symmetric shift operator and investigated its properties. Using the symmetric shift operator, a symmetric differentiation operator is constructed. Conditions, when the symmetric differentiation operator is densely defined, are established. The adjoint operator to differentiation is described. It is shown that in a special case the differentiation operator and the adjoint operator are creation and annihilation operators, that is, they satisfy the canonical commutation relation. Some properties of these operators, related to the symmetric shift, are obtained.