Compact operators in the commutant of essentially normal operators

Abstract

Let $T$ be a bounded, linear operator on a complex, separable, infinite dimensional Hilbert space $H$. We assume that $T$ is an essentially isometric (resp. normal) operator, that is, $I_{H}-T^{\ast }T$ (resp. $TT^{\ast }-T^{\ast }T)$ is compact. For the compactness of $S$ from the commutant of $T,$ some necessary and sufficient conditions are found on $S.$ Some related problems are also discussed.