Operators with powers essentially similar to those of their adjoints

Abstract

Let T be an operator on a Hilbert space H. In the present note the following result is obtained: If T is an operator such that for some integers p , q , S T ∗ p = T q S + K p,q,S{T^{ \ast p}} = {T^q}S + K , where 0 is not in the essential numerical range of S, and K is compact, then for any complex number λ \lambda in the essential spectrum of T, λ ∗ p = λ q {\lambda ^{ \ast p}} = {\lambda ^q} .