Pryimak H. On structures of the set of homomorphisms and functional calculus in algebras of analytic functions on Banach space.

The dissertation describes a set of continuous homomorphisms from H_b(X) into some commutative Banach algebra A. We describes setting conditions for a Banach space X, a commutative Banach algebra A and a homomorphism \Phi:H_b(X)\rightarrow A for which there exists a net (\overline{a}_{\alpha})\subsetA\otimes_\pi X such that operators of functional calculus (\theta_{\overline{a}_{\alpha}}) approximate the homomorphism \Phi on polynomials in H_b(X)or on all functions from H_b(X). The central idea is the application of the Aron-Berner extension to A-valued functional calculus homomorphisms \overline{f}\in H_b(A\bigotimes_{\pi} X,A) to the second dual space. We use the known facts about extensions of analytic functions from H_b(X) to X''. This approach it is possible to use because of establishing the fact that A\bigotimes_{\pi} X'' \subset (A\bigotimes_{\pi} X)'' and A''\bigotimes_{\pi} X\subset (A\bigotimes_{\pi} X)''. We generalized some properties of the translation operator for A-valued analytic functions of bounded type and convolution operations for A-valued homomorphisms. We proved the basic structural theorem for homomorphism \Phi from algebra H_b(X) into some commutative Banach algebra A which can be approximated by homomorphisms of functional calculus in the weak-polynomial topology. The non-classicalA-valued differentiations of algebra H_b(A\bigotimes_{\pi} X,A) are described. It is proved that the operator \overline{\partial}_{(k)}(u_k) is a continuous differentiation on H_b((A\bigotimes _\pi X),A) and proposed formulas of calculation of the algebra-valued derivatives P\in\mathcal{P}(^n(A\bigotimes _\pi X),A) and \overline{f}\in H_b((A\bigotimes _\pi X),A).