Properties which normal operators share with normal derivations and related operators

Abstract

Let S and T be (bounded) scalar operators on a Banach space T and let C(T, S) be the map on &{?), the bounded linear operators on ^ defined by C(, S)(JS) = TX-XS for X in &(?). This paper was motivated by the question: to what extent does C(T f S) behave like a normal operator on Hubert space? It will be shown that C(T, S) does share many of the special properties enjoyed by normal operators. For example it will be shown that the range of C(T, S) meets its null space at a positive angle and that C(T,S) is Hermitian if T and S are Hermitian. However, if f is a Hubert space then C(T, S) is a spectral operator if and only if the spectrum of T and the spectrum of S are both finite.