Similarity and approximation of EP operators

Abstract

A Hilbert space operator T is called an EP operator if its range R(T) is closed and R(T)=R(T⁎). First, we show that the class of invertible operators and the class of EP operators have the same norm closure and the same interior. Second, we show that the similarity orbit of an nonzero operator T is contained in the class of EP operators if and only if T is invertible. Finally, we describe when Toeplitz operators and weighted shifts possess the EP property. We obtain similar results for an operator property weaker than the EP property.