Properties which normal operators share with normal derivations and related operators

Abstract

Let $S$ and $T$ be (bounded) scalar operators on a Banach space $\scr X$ and let $C(T,S)$ be the map on $\scr B(\scr X)$, the bounded linear operators on $\scr X$, defined by $C(T,S)(X)=TX-XS$ for $X$ in $\scr B(\scr X)$. This paper was motivated by the question: to what extent does $C(T,S)$ behave like a normal operator on Hilbert space? It will be shown that $C(T,S)$ does share many of the special properties enjoyed by normal operators. For example, it is 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 $\scr X$ is a Hilbert 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.