PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 8, Pages 2303–2314 S 0002-9939(99)04802-9 Article electronically published on April 8, 1999 INTEGRAL REPRESENTATION FORMULA FOR GENERALIZED NORMAL DERIVATIONS DANKO R. JOCI ´ C (Communicated by Palle E. T. Jorgensen) Abstract. For generalized normal derivations, acting on the space of all bounded Hilbert space operators, the following integral representation for- mulas hold: f (A)X − Xf (B)=  σ(A)  σ(B) f (z) − f (w) z − w E(dz)(AX − XB)F (dw), (1) and ‖f (A)X − Xf (B)‖ 2 2 =  σ(A)  σ(B)     f (z) − f (w) z − w     2 ‖E(dz)(AX − XB)F (dw)‖ 2 2 , (2) whenever AX − XB is a Hilbert-Schmidt class operator and f is a Lipschitz class function on σ(A) ∪ σ(B). Applying this formula, we extend the Fuglede- Putnam theorem concerning commutativity modulo Hilbert-Schmidt class, as well as trace inequalities for covariance matrices of Muir and Wong. Some new monotone matrix functions and norm inequalities are also derived. 1. Introduction In his paper [Ki] Kittaneh proved the following theorem for bounded Hilbert space operators: Theorem 1.1. Let A and B be normal bounded operators such that AX − XB belongs to the Hilbert-Schmidt class for some bounded operator X, and let f be a Lipschitz function defined on σ(A) ∪ σ(B). Then f (A)X − Xf (B) also belongs to the Hilbert-Schmidt class; moreover, ‖f (A)X − Xf (B)‖ 2 ≤ L‖AX − XB‖ 2 , (3) where L stands for the Lipschitz constant associated to f. The proof presented there is based on a famous Voiculescu theorem representing every normal operator as a sum of one diagonal and one Hilbert-Schmidt class operator which has the arbitrary small Hilbert-Schmidt norm (see [Vo]). If A = B is self-adjoint, then (3) holds for arbitrary closed derivations, as shown recently in Received by the editors January 2, 1997 and, in revised form, October 28, 1997. 1991 Mathematics Subject Classification. Primary 47A13, 47B10, 47B15, 47B47, 47B49; Sec- ondary 47A30, 47A60. Key words and phrases. Double operator integrals, unitarily invariant norms, Ky-Fan domi- nance property. c 1999 American Mathematical Society 2303