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 - Xƒ(B)∥2 ≤ L∥AX - XB∥2 for ƒ belongs to a certain class of functions. Furthermore, we investigate the similar problem in the case that S, T are hyponormal operators and X ϵ ℒ(ℋ) is such that SX - XT belongs to a norm ideal (J, ∥·∥J), and we prove that ƒ(S)X - Xƒ(T) ϵ J and ∥f(S)X - Xƒ(T)∥J ≤ C∥SX - XT∥J for ƒ being in a certain class of functions. [ABSTRACT FROM AUTHOR]