On the norm-closure of the class of hypercyclic operators

Abstract

Let T be a bounded linear operator acting on a complex, separable, infinitedimensional Hilbert space and let f : D → C be an analytic function defined on an open set D ⊆ C which contains the spectrum of T . If T is the limit of hypercyclic operators and if f is nonconstant on every connected component of D, then f(T ) is the limit of hypercyclic operators if and only if f(σW(T )) ∪ {z ∈ C : |z| = 1} is connected, where σW(T ) denotes the Weyl spectrum of T . 1. Terminology and introduction. In this note X always denotes a complex, infinite-dimensional Banach space and L(X) the Banach algebra of all bounded linear operators on X. We write K(X) for the ideal of all compact operators on X. For T ∈ L(X) the spectrum of T is denoted by σ(T ). The reader is referred to [5] for the definitions and properties of Fredholm operators, semi-Fredholm operators and the index ind(T ) of a semi-Fredholm operator T in L(X). For T ∈ L(X) we will use the following notations: %F (T ) = {λ ∈ C : λI − T is Fredholm}, %s-F(T ) = {λ ∈ C : λI − T is semi-Fredholm}, %W(T ) = {λ ∈ %F(T ) : ind(λI − T ) = 0}, σ0(T ) = {λ ∈ σ(T ) : λ is isolated in σ(T ), and λ ∈ %F(T )}, σF(T ) = C %F(T ), σs-F(T ) = C %s-F(T ), σW(T ) = C %W(T ) (Weyl spectrum), Hol(T ) = {f : D(f)→ C : D(f) is open, σ(T ) ⊆ D(f), f is holomorphic}, Hol(T ) = {f ∈ Hol(T ) : f is nonconstant on every connected component of D(f)}. 1991 Mathematics Subject Classification: Primary 47B99.