Hereditarily hypercyclicity and supercyclicity of weighted shifts

Yu-Xia Liang; Ze-Hua Zhou

doi:10.4134/JKMS.2014.51.2.363

Journal of the
Korean Mathematical Society JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2014; 51(2): 363-382

Printed March 1, 2014

https://doi.org/10.4134/JKMS.2014.51.2.363

Hereditarily hypercyclicity and supercyclicity of weighted shifts

Yu-Xia Liang and Ze-Hua Zhou

Tianjin University, Tianjin University

Abstract

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Abstract

In this paper we first characterize the hereditarily hypercyclicity of the unilateral (or bilateral) weighted shifts on the spaces $L^2(\mathbb{N},\mathcal{K})\; (\mbox{or}\;L^2(\mathbb{Z},\mathcal{K}))$ with weight sequence $\{A_n\} $ of positive invertible diagonal operators on a separable complex Hilbert space $\mathcal{K}.$ Then we give the necessary and sufficient conditions for the supercyclicity of those weighted shifts, which extends some previous results of H. Salas. At last, we give some conditions for the supercyclicity of three different weighted shifts.

Keywords: hereditarily hypercyclic, supercyclic, weighted shifts

MSC numbers: 47A16, 47B37