On Lipschitz function of subnormal operators

We prove the following:Theorem: Let S be a subnormal operator on an infinite dimensional separable complex Hilbert space H. Let f be analytic and Lipschitz on a neighborhood e of the spectrum σ(S) of S. Then ||f(S)X-X f(S)||2 ≤ K||Sx-X S||2 for some K> O and for every X ε B(H), the algebra of all bounded linear operators on H, where ||.||2 is the Hilbert-Schmidt norm. A.M.S. subject classification number 1990: 47 B 20, 47 A 25. (authors). 7 refs