Dynamics of non-convolution operators and holomorphy types

In this article we study the hypercyclic behavior of non-convolution operators defined on spaces of analytic functions of different holomorphy types over Banach spaces. The operators in the family we analyze are a composition of differentiation and composition operators, and are extensions of operat...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2018-12, Vol.468 (2), p.622-641
Main Authors: Muro, Santiago, Pinasco, Damián, Savransky, Martín
Format: Article
Language:English
Subjects:
Composition operators
Differentiation operators
Holomorphy types
Hypercyclic operators
Non-convolution operators
Strongly mixing operators
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Summary:In this article we study the hypercyclic behavior of non-convolution operators defined on spaces of analytic functions of different holomorphy types over Banach spaces. The operators in the family we analyze are a composition of differentiation and composition operators, and are extensions of operators in H(C) studied by Aron and Markose in 2004. The dynamics of this class of operators, in the context of one and several complex variables, was further investigated by many authors. It turns out that the situation is somewhat different and that some purely infinite dimensional difficulties appear. For example, in contrast to the several complex variable case, it may happen that the symbol of the composition operator has no fixed points and still, the operator is not hypercyclic. We also prove a Runge type theorem for holomorphy types on Banach spaces.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2018.08.017