a-Weyl’s theorem and hypercyclicity

Abstract

Let H be a complex infinite dimensional Hilbert space, B(H) be the algebra of all bounded linear operators acting on H, and \(\overline{HC(H)}\) \((\overline{SC(H)})\) be the norm closure of the class of all hypercyclic operators (supercyclic operators) in B(H). An operator \(T\in B(H)\) is said to be with hypercyclicity (supercyclicity) if T is in \(\overline{HC(H)}\) \((\overline{SC(H)})\). Using a new spectrum defined from “consistent in invertibility”, this paper gives necessary and sufficient conditions that T is with a-Browder’s theorem or with a-Weyl’s theorem. Further, this paper gives a necessary and sufficient condition that T is a-isoloid, with a-Weyl’s theorem and with hypercyclicity (supercyclicity) concurrently. Also, the relations between that T is with hypercyclicity (supercyclicity) and that T is both with a-Weyl’s theorem and a-isoloid are discussed by means of the new spectrum.