Common hypercyclic vectors for the conjugate class of a hypercyclic operator |
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Authors: | Kit C Chan |
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Institution: | a Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OH 43403, United States b Department of Mathematics, Statistics, and Computer Sciences, Marquette University, Milwaukee, WI 53201, United States |
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Abstract: | Given a separable, infinite dimensional Hilbert space, it was recently shown by the authors that there is a path of chaotic operators, which is dense in the operator algebra with the strong operator topology, and along which every operator has the exact same dense Gδ set of hypercyclic vectors. In the present work, we show that the conjugate set of any hypercyclic operator on a separable, infinite dimensional Banach space always contains a path of operators which is dense with the strong operator topology, and yet the set of common hypercyclic vectors for the entire path is a dense Gδ set. As a corollary, the hypercyclic operators on such a Banach space form a connected subset of the operator algebra with the strong operator topology. |
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Keywords: | Hypercyclic operator Common hypercyclic vectors Paths of operators Strong operator topology Denseness _method=retrieve& _eid=1-s2 0-S0022247X1000661X& _mathId=si3 gif& _pii=S0022247X1000661X& _issn=0022247X& _acct=C000051805& _version=1& _userid=1154080& md5=6fc313657c4ed3dc957bd355ad52b0bf')" style="cursor:pointer Gδ set" target="_blank">" alt="Click to view the MathML source" title="Click to view the MathML source">Gδ set |
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