Common Hypercyclic Vectors for the Unitary Orbit of a Hypercyclic Operator
Journal of Mathematical Analysis and Applications
For a separable, infinite dimensional Hilbert space, it was recently shown by the authors that the similarity orbit of a hypercyclic operator contains a path of operators which is dense in the operator algebra with the strong operator topology, and yet the set of common hypercyclic vectors for the entire path is a dense Gδ set. Motivated by that result, we show in the present paper that the unitary orbit of any hypercyclic operator contains a path of operators whose closure contains the entire unitary orbit with the strong operator topology, and yet every nonzero vector in the linear span of the orbit of a given hypercyclic vector is a common hypercyclic vector for the entire path.
Chan, Kit C. and Sanders, Rebecca, "Common Hypercyclic Vectors for the Unitary Orbit of a Hypercyclic Operator" (2012). Mathematics, Statistics and Computer Science Faculty Research and Publications. 593.