Hypercyclic operators on algebra of symmetric analytic functions on $\ell_p$

In the paper, it is proposed a method of construction of hypercyclic composition operators on $H(\mathbb{C}^n)$ using polynomial automorphisms of $\mathbb{C}^n$ and symmetric analytic functions on $\ell_p.$ In particular, we show that an "symmetric translation" operator is hypercyclic on a Frechet algebra of symmetric entire functions on $\ell_p$ which are bounded on bounded subsets.

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Polynomial automorphisms and hypercyclic operators on spaces of analytic functions

Archiv der Mathematik ◽

10.1007/s00013-007-2043-4 ◽

2007 ◽

Vol 89 (2) ◽

pp. 157-166 ◽

Cited By ~ 3

Author(s):

Zoryana Novosad ◽

Andriy Zagorodnyuk

Keyword(s):

Analytic Functions ◽

Hypercyclic Operators ◽

Polynomial Automorphisms ◽

Spaces Of Analytic Functions

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Algebras of symmetric analytic functions on Cartesian powers of Lebesgue integrable in a power $p\in [1,+\infty)$ functions

Carpathian Mathematical Publications ◽

10.15330/cmp.13.2.340-351 ◽

2021 ◽

Vol 13 (2) ◽

pp. 340-351

Author(s):

T.V. Vasylyshyn

Keyword(s):

Banach Space ◽

Analytic Functions ◽

Entire Functions ◽

Complex Banach Space ◽

Cartesian Power ◽

Entire Complex ◽

Continuous Complex ◽

Fréchet Algebra ◽

Algebraic Basis ◽

Complex Valued

The work is devoted to the study of Fréchet algebras of symmetric (invariant under the composition of every of components of its argument with any measure preserving bijection of the domain of components of the argument) analytic functions on Cartesian powers of complex Banach spaces of Lebesgue integrable in a power $p\in [1,+\infty)$ complex-valued functions on the segment $[0,1]$ and on the semi-axis. We show that the Fréchet algebra of all symmetric analytic entire complex-valued functions of bounded type on the $n$th Cartesian power of the complex Banach space $L_p[0,1]$ of all Lebesgue integrable in a power $p\in [1,+\infty)$ complex-valued functions on the segment $[0,1]$ is isomorphic to the Fréchet algebra of all analytic entire functions on $\mathbb C^m,$ where $m$ is the cardinality of the algebraic basis of the algebra of all symmetric continuous complex-valued polynomials on this Cartesian power. The analogical result for the Fréchet algebra of all symmetric analytic entire complex-valued functions of bounded type on the $n$th Cartesian power of the complex Banach space $L_p[0,+\infty)$ of all Lebesgue integrable in a power $p\in [1,+\infty)$ complex-valued functions on the semi-axis $[0,+\infty)$ is proved.

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Hypercyclic Behavior of Translation Operators on Spaces of Analytic Functions on Hilbert Spaces

Journal of Function Spaces ◽

10.1155/2015/139289 ◽

2015 ◽

Vol 2015 ◽

pp. 1-8 ◽

Cited By ~ 1

Author(s):

Zoryana Mozhyrovska ◽

Andriy V. Zagorodnyuk

Keyword(s):

Hilbert Spaces ◽

Analytic Functions ◽

Composition Operators ◽

Translation Operator ◽

Translation Operators ◽

Spaces Of Analytic Functions

We consider special Hilbert spaces of analytic functions of many infinite variables and examine composition operators on these spaces. In particular, we prove that under some conditions a translation operator is bounded and hypercyclic.

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Entire Analytic Functions of Unbounded Type on Banach Spaces and Their Lineability

Axioms ◽

10.3390/axioms10030150 ◽

2021 ◽

Vol 10 (3) ◽

pp. 150

Author(s):

Andriy Zagorodnyuk ◽

Anna Hihliuk

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Analytic Functions ◽

Entire Functions ◽

Dimensional Algebra ◽

Dimensional Banach Space ◽

Infinite Dimensional ◽

Linear Subspaces ◽

Infinite Dimensional Banach Space

In the paper we establish some conditions under which a given sequence of polynomials on a Banach space X supports entire functions of unbounded type, and construct some counter examples. We show that if X is an infinite dimensional Banach space, then the set of entire functions of unbounded type can be represented as a union of infinite dimensional linear subspaces (without the origin). Moreover, we show that for some cases, the set of entire functions of unbounded type generated by a given sequence of polynomials contains an infinite dimensional algebra (without the origin). Some applications for symmetric analytic functions on Banach spaces are obtained.

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Essential Norm and Weak Compactness of Composition Operators on Weighted Banach Spaces of Analytic Functions

Canadian Mathematical Bulletin ◽

10.4153/cmb-1999-016-x ◽

1999 ◽

Vol 42 (2) ◽

pp. 139-148 ◽

Cited By ~ 59

Author(s):

José Bonet ◽

Paweł Dománski ◽

Mikael Lindström

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Analytic Functions ◽

Composition Operators ◽

Essential Norm ◽

Weak Compactness ◽

Weakly Compact ◽

Point Evaluation ◽

Weighted Banach Spaces ◽

Spaces Of Analytic Functions

AbstractEvery weakly compact composition operator between weighted Banach spaces of analytic functions with weighted sup-norms is compact. Lower and upper estimates of the essential norm of continuous composition operators are obtained. The norms of the point evaluation functionals on the Banach space are also estimated, thus permitting to get new characterizations of compact composition operators between these spaces.

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Weak-Star Continuous Analytic Functions

Canadian Journal of Mathematics ◽

10.4153/cjm-1995-035-1 ◽

1995 ◽

Vol 47 (4) ◽

pp. 673-683 ◽

Cited By ~ 27

Author(s):

R. M. Aron ◽

B. J. Cole ◽

T. W. Gamelin

Keyword(s):

Unit Ball ◽

Finite Type ◽

Analytic Functions ◽

Approximation Property ◽

Entire Functions ◽

Open Unit ◽

Closed Unit Ball ◽

Open Unit Ball ◽

Weakly Continuous ◽

Weak Star

AbstractLet 𝒳 be a complex Banach space, with open unit ball B. We consider the algebra of analytic functions on B that are weakly continuous and that are uniformly continuous with respect to the norm. We show these are precisely the analytic functions on B that extend to be weak-star continuous on the closed unit ball of 𝒳**. If 𝒳* has the approximation property, then any such function is approximable uniformly on B by finite polynomials in elements of 𝒳*. On the other hand, there exist Banach spaces for which these finite-type polynomials fail to approximate. We consider also the approximation of entire functions by finite-type polynomials. Assuming 𝒳* has the approximation property, we show that entire functions are approximable uniformly on bounded sets if and only if the spectrum of the algebra of entire functions coincides (as a point set) with 𝒳**.

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