LIMITS OF HYPERCYCLIC AND SUPERCYCLIC OPERATORS

An operator T on a complex, separable, infinite dimensional Hilbert space H is hypercyclic if there is a vector y in H such that the orbit {y, Ty, T2y, ...} is dense in H. It is shown that if T is hypercyclic, then T* does not have any eigenvalue and that the union of the spectrum of T and the unit circle is a connected set. This result is used to obtain a spectral characterization of the norm-closure of the class HC(H) of all hypercyclic operators acting on H, as well as a formula for the distance from a given operator to HC(H). Analogous results are obtained for the closely related class SC(H) of all supercyclic operators. This paper includes some information about the structure of the set of all hypercyclic (supercyclic) vectors of a given hypercyclic (supercyclic, resp.) operator T. © 1991.