Taras O. Algebraic and topological structures on spectra of analytic functions of Banach spaces.

The thesis is devoted to algebras of analytic functions on Banach spaces investigationsand analysis of algebraic and topological structures on their spectra. Some main properties of the algebras of block-diagonal analytic functions on Banach spaces and their place in the general theory were described by establishing connectedness between algebras of block-diagonal analytic functions and algebras of analytic functionsof bounded type. For certain partial cases of algebras of block-diagonal analytic functions on Banach spaces l1,l2 it were described their spectra,somehomomorphismson algebrasof block-diagonal analytic functions on Banach spaces were investigated.The "multiplicative" convolution was constructed on the space of all continuous linear functionals on algebra of analytic functions of bounded type which actually is a generalization of Arens extension of the operation of multiplying in Banach algebra Ato it's bidual space A**. Some main properties and continuity of this convolution were proved. The general convolution rules for fixed two characters with different representation which belongs to spectrum of algebraanalytic function of bounded type were established. In the general case it was showed that for "additive" and "multiplicative" convolutions the distributive rule does not hold. It was proved that the set of all invertible relatively "multiplicative" convolution elements of the spectrum of algebra of analytic functions of bounded type is analytic manifolds with respect to some natural topology.