Polynomial automorphisms and hypercyclic operators on spaces of analytic functions

Abstract

We consider hypercyclic composition operators on H(C n ) which can be obtained from the translation operator using polynomial automorphisms of C n . In particular we show that if CS is a hypercyclic operator for an affine automorphism S on H(C n ), then S = Θ • (I + b) • Θ −1 + a for some polynomial automorphism Θ and vectors a and b, where I is the identity operator. Finally, we prove the hypercyclicity of ''symmetric translations'' on a space of symmetric analytic functions on 1.

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

“…Aron and Bès in [7] proved that the operator of composition with translation is hypercyclic in the space of weakly continuous analytic functions on all bounded subsets of a separable Banach space which are bounded on bounded subsets. Hypercyclic composition operators on spaces of analytic functions of finite and infinite many variables were studied also in [8]. In [9] Chan and Shapiro show that is hypercyclic in various Hilbert spaces of entire functions on C. More detailed, they considered Hilbert spaces of entire functions of one complex variable…”

Analytic Automorphisms and Transitivity of Analytic Mappings

“…It is well known that the translation operator f (x) → f (x + a) defined on the space of entire functions H(C n ) is hypercyclic if a = 0. Infinite-dimensional generalizations of this results can be found in [13], [8], [15].…”

“…Analytic automorphisms can be applied, also, for linear dynamics. If C Φ is a hypercyclic composition linear operator on the space of all analytic functions H(X) on a topological vector space X and F is an analytic automorphism of X, then C F•Φ•F −1 is a hypercyclic composition linear operator which does not commute, in general, with C Φ [8]. Note that the hypercyclisity of C Φ on H(C) for Φ : x → x + a was proved by Birkhoff in [9] and generalized in [10], [11], [12], for the case H(C n ) and for some infinite dimensional cases in [13], [14], [15], [8].…”

Hypercyclic operators on spaces of block-symmetric analytic functions

“…Note that an analog of the Godefroy-Shapiro Theorem for weakly continuous analytic functions on Banach spaces which are bounded on bounded subsets was proved by Aron and Bés in [1]. A symmetric version of the Godefroy-Shapiro Theorem is proved in [9]. The purpose of this paper is to extend the results on hypercyclicity for the case of a special translation operator on the space of block-symmetric analytic functions of bounded type on ℓ 1 .…”