Some Estimates of Certain Subnormal and Hyponormal Derivations

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.