Kravtsiv V. Symmetric analytic functions on products of Banach spaces.

In the thesis the analytic functions on the finite and infinite products of Banach spaces with the symmetric bases, which are invariant with respect to actions of some natural subgroup of the group of isometric operators, are investigated.Algebraic bases of the algebra of block-symmetric polynomials on the infinite products of Banach space are described; the algebraic dependences between generate elements in the finite dimensional case are shown.An analogue of the Hilbert Nullstellensatz for ideals, generated by block-symmetric polynomials is proved; an analogue of Martin's formula with operator of symmetric shift is established; an analogue of Newton's formula for respective systems of generators of block-symmetric polynomials algebra is established. It is described the spectra of algebras of block-symmetric analytic functions of bounded type on the -sum of by exponential type functions of two complex variables.