A Note on Subnormal and Hyponormal Derivations

Abstract

In this note we prove that if A and B are subnormal operators and X is a bounded linear operator such that AX XB is a Hilbert-Schmidt operator, then f(A)X Xf(B) is also a Hilbert-Schmidt operator and ||f(A)X Xf(B)||2 L||AX XB||2, for f belonging to a certain class of functions. Furthermore, we investigate the similar problem in the case that S, T are hyponormal operators and X 2 L(H) is such that SX XT belongs to a norm ideal (J, || · ||J) and prove that f(S)X Xf(T) 2 J and ||f(S)X Xf(T)||J C ||SX XT||J, for f in a certain class of functions.