Strongly Mixing Convolution Operators on Fréchet Spaces of Holomorphic Functions

Abstract

Abstract. A theorem of Godefroy and Shapiro states that non-trivial convolution operators on the space of entire functions on C n are hypercyclic. Moreover, it was shown by Bonilla and Grosse-Erdmann that they have frequently hypercyclic functions of exponential growth. On the other hand, in the infinite dimensional setting, the Godefroy-Shapiro theorem has been extended to several spaces of entire functions defined on Banach spaces. We prove that on all these spaces, non-trivial convolution operators are strongly mixing with respect to a gaussian probability measure of full support. For the proof we combine the results previously mentioned and we use techniques recently developed by Bayart and Matheron. We also obtain the existence of frequently hypercyclic entire functions of exponential growth.

Duality results in Banach and quasi-Banach spaces of homogeneous polynomials and applications

“…For the theory of hypercyclic operators and its ramifications we refer to [2,4,32] and references therein. We remark that several results on the hypercyclicity of operators on spaces of entire functions on infinitely many complex variables appeared later (see, e.g., [3,5,6,12,13,30,32,49,52]). In 2007, Carando, Dimant and Muro [12] proved some general results, including a solution to a problem posed in [1], that encompass as particular cases several of the above mentioned results.…”

“…and it follows that T A(·) k ∈ P N,((r,q);(s,p)) ( n−k E). Moreover [49] shows that nontrivial convolution operators on certain spaces of entire functions on a Banach space are strongly mixing in the gaussian sense, in particular frequently hypercyclic. We believe that, using the corresponding auxiliary results from [5], their result holds for H Θb (E), when Θ is a π 1 -holomorphy type.…”

Dynamics of non-convolution operators and holomorphy types

“…There is a natural way to associate to a holomorphy type A spaces of holomorphic functions of bounded type on a Banach space E, namely the holomorphic functions that have a given A-radius of convergence at each point of E (see for example [11,16,18,35]…”

“…The Godefroy -Shapiro theorem has been improved by Bonilla and Grosse-Erdmann in [9]. They showed that non-trivial convolution operators are frequently hypercyclic (see also [21,35]).…”

Hypercyclicity, existence and approximation results for convolution operators on spaces of entire functions

“…For the theory of hypercyclic operators and its ramifications we refer to [2,29,30]. We remark that several results on the hypercyclicity of operators on spaces of entire functions in infinitely many complex variables appeared later (see, e.g., [1,3,4,5,9,10,25,27,29,50,53]). In 2007, Carando, Dimant and Muro [9] proved some general results that encompass as particular cases several of the above mentioned results.…”