On some perturbation inequalities for operators (Q1112306)

Let E, F be any two Hilbert spaces, and let L(E,F) denote the space of bounded linear operators from E to F. Let L(E,E) be denoted simply as L(E). It has long been known [see \textit{M. Rosenblum}, Duke Math. J. 23, 263-269 (1956; Zbl 0073.330)] that if A and B are elements of L(E) and L(F), respectively, such that spectrum of A and spectrum of B are disjoint, then for every S in L(F,E) the equation \(AX-XB=S\) has a unique solution X in L(F,E). An estimate for the norm of the solution obtained by \textit{R. Bhatia}, \textit{C. Davis} and \textit{A. McIntosh} for normal operators A and B [Linear Algebra Appl. 52-53, 45-67 (1983; Zbl 0518.47013)] is shown to be valid for a larger class (for subnormal operators). Recall that an operator A in L(E) is subnormal if it has a normal extension; i.e., there exist a Hilbert space K containing H as a subspace and a normal operator M in L(K) which leaves H invariant and which coincides with A when restricted to H.