Journal of Operator Theory

Volume 61, Issue 2, Spring 2009 pp. 369-380.

A hypercyclic operator whose direct sum $T \oplus T$ is not hypercyclic

Authors: Manuel de la Rosa (1) and Charles Read (2)
Author institution: (1) Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, UK
(2) Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, UK

Summary: The present article answers in the negative the great and long-standing problem in hypercyclicity posed by D. Herrero: \textit{Is $T \oplus T$ hypercyclic whenever $T$ is?} It also answers simultaneously the significant question asked by J. Bes, A. Peris, F. Leon-Saavedra and A. Montes-Rodriguez: \textit{Does every hypercyclic operator satisfy the Hypercyclicity Criterion?}

Keywords: Hypercyclic operators, hypercyclicity criterion, cyclic vectors, direct sums of hypercyclic operators, Banach spaces.

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