Some estimates of certain subnormal and hyponormal derivations (Q938512)

Summary: 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 \(\|f(A)X-Xf(B)\|_2\leq L\|AX-XB\|_2\) 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 \(X\in{\mathcal L}({\mathcal H})\) is such that \(SX-XT\) belongs to a norm ideal \((J,\|\cdot\|_J)\), and we prove that \(f(S)X-Xf(T)\in J\) and \(\|f(S)X-Xf(T)\|_J\leq C\|SX-XT\|_J\) for \(f\) being in a certain class of functions.