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

Українська версія

Thesis for the degree of Candidate of Sciences (CSc)

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  • 01.01.01 - Математичний аналіз


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К 20.051.09

Kolomyia Educational-Scientific Institute The Vasyl Stefanyk Precarpathian National University


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).


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