International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 362409, 6 pages
doi:10.1155/2008/362409

Some estimates of certain subnormal and hyponormal derivations

Abstract

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 belongs to a certain class of functions. Furthermore, we investigate the similar problem in the case that S, T are hyponormal operators and X∈ℒ(ℋ) is such that SX−XT belongs to a norm ideal (J,‖⋅‖J), and we prove that f(S)X−Xf(T)∈J and ‖f(S)X−Xf(T)‖J≤C‖SX−XT‖J for f being in a certain class of functions.