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Journal of Operator Theory

Volume 66, Issue 1, Summer 2011  pp. 107-124.

An SOT-dense path of chaotic operators with same hypercyclic vectors

Authors Kit C. Chan (1) and Rebecca Sanders (2)
Author institution: (1) Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, 43403, U.S.A.
(2) Department of Mathematics, Statistics, and Computer Science, Marquette University, Milwaukee, 53201, U.S.A.

Summary:  Recently many authors have obtained interesting results on the existence of a dense $G_\delta$ set of common hypercyclic vectors for a path of operators. We show that on a separable infinite dimensional Hilbert space, there is a path of chaotic operators that is dense in the operator algebra with the strong operator topology, and yet each operator along the path has the exact same dense $G_{\delta}$ set of hypercyclic vectors. As a corollary, the operators having that particular set of hypercyclic vectors form a connected subset of the operator algebra with the strong operator topology.

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