On Normal Derivations

Abstract

Let AT be the derivation on SBpf) defined by Ar(A')= TX-XT (T,Xe 93(^)).We prove that if T is an isometry or a normal operator, then the range of AT is orthogonal to the null space of Aj,.Also, we prove that if T is normal with an infinite number of points in its spectrum then the closed linear span of the range and the null space of Ar is not all of S8(^f).