Countably Generated Algebras of Analytic Functions on Banach Spaces

Abstract

In the paper, we study various countably generated algebras of entire analytic functions on complex Banach spaces and their homomorphisms. Countably generated algebras often appear as algebras of symmetric analytic functions on Banach spaces with respect to a group of symmetries, and are interesting for their possible applications. Some conditions of the existence of topological isomorphisms between such algebras are obtained. We construct a class of countably generated algebras, where all normalized algebraic bases are equivalent. On the other hand, we find non-isomorphic classes of such algebras. In addition, we establish the conditions of the hypercyclicity of derivations in countably generated algebras of entire analytic functions of the bounded type. We use methods from the theory of analytic functions of several variables, the theory of commutative Fréchet algebras, and the theory of linear dynamical systems.

Analytic Invariants of Semidirect Products of Symmetric Groups on Banach Spaces

“…In the general case, the functional δ z does not exhaust the spectrum of H b (X), and it may have a complicated structure. It was a motivation for studying the spectra of the countable generated subalgebras of H b (X), in particular, the subalgebras of symmetric functions (see, e.g., [10]).…”

“…Algebras of symmetric analytic functions of a bounded type on p were considered in [8,9]. These investigations were continued in a number of papers (see, e.g., [10] and the references therein). A continual group of symmetry and the corresponding algebras of symmetric analytic functions on L ∞ were investigated in [11][12][13].…”

“…Thus, the first important question concerning an algebra of symmetric polynomials is about the existence of a countable algebraic basis (or a generating sequence) of polynomials. The algebras of analytic functions on X, generated by a countable family of polynomials were systematically studied in [10,[14][15][16].…”

The Backward Shift and Two Infinite-Dimension Phenomena in Banach Spaces

“…form a linear basis in the linear space of n-homogeneous subsymmetric polynomials on p [32]. More information about spaces and algebras generated by symmetric and subsymmetric polynomials can be found in [33][34][35][36][37] and the references therein. It is easy to see that if P is symmetric, then it is subsymmetric, and the inverse statement is not true.…”