Some Estimates of Certain Subnormal and Hyponormal Derivations

Abstract EN

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.