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Hypercyclicity of composition operators in Stein manifolds

Author: Sylwester Zajac
Journal: Proc. Amer. Math. Soc. 144 (2016), 3991-4000
MSC (2010): Primary 47B33; Secondary 32H50
Published electronically: March 17, 2016
MathSciNet review: 3513554
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Abstract: We characterize hypercyclic composition operators $ C_\varphi :f\mapsto f\circ \varphi $ on the space of holomorphic functions on a connected Stein manifold $ \Omega $ with $ \varphi $ being a holomorphic self-map of $ \Omega $.

In turns out that in the case when all balls with respect to the Carathéodory pseudodistance are relatively compact in $ \Omega $, a much simpler characterization may be obtained (many natural classes of domains in $ \mathbb{C}^N$ satisfy this condition). Moreover, we show that in such a class of manifolds, as well as in simply connected and infinitely connected planar domains, hypercyclicity of $ C_\varphi $ implies its hereditary hypercyclicity with respect to the full sequence of natural numbers.

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Sylwester Zajac
Affiliation: Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland

Keywords: Spaces of holomorphic functions, composition operator, hypercyclic operator, holomorphic convexity, $\OO(\Omega)$-convexity.
Received by editor(s): May 22, 2015
Received by editor(s) in revised form: October 14, 2015, and November 21, 2015
Published electronically: March 17, 2016
Communicated by: Franc Forstneric
Article copyright: © Copyright 2016 American Mathematical Society

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