Spectra of algebras of block-symmetric analytic functions of bounded type

Zagorodnyuk, Andriy; Kravtsiv, V. V.

doi:10.30970/ms.58.1.69-81

Cited by 4 publications

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References 34 publications

(37 reference statements)

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“…Often, S-symmetric polynomials on X 1 × • • • × X s are called block-symmetric polynomials. Algebras of block-symmetric analytic functions were studied in [22,25,32,[61][62][63] for the case when X j = p j and S j are groups of permutations of basis vectors in p j , and in [20,31] for the case when X j = L ∞ (Ω j ) and S j are groups of measurable, measure-preserving automorphisms of measure spaces Ω j .…”

Section: Proof the Restriction Of Cmentioning

confidence: 99%

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Countably Generated Algebras of Analytic Functions on Banach Spaces

Novosad,

Vasylyshyn,

Zagorodnyuk

2023

Axioms

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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.

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Section: Proof the Restriction Of Cmentioning

confidence: 99%

“…These investigations were continued in [17,18] and in [19,20] for the case X = L p . Various results in this direction for different subalgebras were obtained in [21][22][23][24][25][26][27][28][29][30][31][32].…”

Section: Introductionmentioning

confidence: 99%

Countably Generated Algebras of Analytic Functions on Banach Spaces

Novosad,

Vasylyshyn,

Zagorodnyuk

2023

Axioms

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Symmetric linear functionals on the Banach space generated by pseudometrics

Nykorovych,

Vasylyshyn

2024

Mat. Stud.

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In this work we consider the notion of $B$-equivalence of pseudometrics.Two pseudometrics $d_1$ and $d_2$ on a set $X$ are called $B$-equivalent, where $B$ is a subgroup of the group of all bijections on $X,$ if there exists an element $b$ of $B$ such that $d_1(x,y) = d_2(b(x),b(y))$ for every $x,y\in X,$ that is, $d_1$ can be obtained from $d_2$ by permutating elements of $X$ with the aid of the bijection $b.$The group $B$ generates the group $\widehat B$ of transformations of the set of all pseudometricson $X,$ elements of which act as $d(\cdot, \cdot)\mapsto d(b(\cdot),b(\cdot)),$ where $d$ is a pseudometrics on $X$ and $b\in B.$ A function $f$ on the set of all pseudometrics on $X$is called $\widehat B$-symmetric if $f$ is invariant under the action on its argument of elements of the group $\widehat B.$If two pseudometrics $d_1$ and $d_2$ are $B$-equivalent, then $f(d_1)=f(d_2)$ for every $\widehat B$-symmetric function $f.$ In general, the technique of symmetric functions is well-developed for the case of symmetric continuous polynomials and, in particular, for the case of symmetric continuous linear functionals on Banach spaces. To use this technique for the construction of $\widehat B$-symmetricfunctions on sets of pseudometrics, we map these sets to some appropriate Banach space $V$, which is isometrically isomorphic to the Banach space $\ell_1$of all absolutely summing real sequences. Weinvestigate symmetric (with respect to an arbitrary group of symmetry, elements of whichmap the standard Schauder basis of $\ell_1$ into itself) linear continuous functionalson $\ell_1.$ We obtain the complete description of the structure of these functionals.Also we establish analogical results for symmetric linear continuous functionals on the space $V.$ These results are used for the construction of $\widehat B$-symmetric functionals on the set of all pseudometrics on an arbitrary set $X$ for the following case:the group $B$ of bijections on $X,$ that generates the group $\widehat B,$ is such that the set of all $x\in X,$ for which there exists $b\in B$ such that $b(x)\neq x,$is finite.

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Analytic Invariants of Semidirect Products of Symmetric Groups on Banach Spaces

Baziv,

Zagorodnyuk

2023

Symmetry

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We consider algebras of polynomials and analytic functions that are invariant with respect to semidirect products of groups of bounded operators on Banach spaces with symmetric bases. In particular, we consider algebras of so-called block-symmetric and double-symmetric analytic functions on Banach spaces ℓp(Cn) and the homomorphisms of these algebras. In addition, we describe an algebraic basis in the algebra of double-symmetric polynomials and discuss a structure of the spectrum of the algebra of double-symmetric analytic functions on ℓp(Cn).

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Spectra of algebras of block-symmetric analytic functions of bounded type

Cited by 4 publications

References 34 publications

Countably Generated Algebras of Analytic Functions on Banach Spaces

Countably Generated Algebras of Analytic Functions on Banach Spaces

Symmetric linear functionals on the Banach space generated by pseudometrics

Analytic Invariants of Semidirect Products of Symmetric Groups on Banach Spaces

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