Integral representation formula for generalized normal derivations

Abstract

For generalized normal derivations, acting on the space of all bounded Hubert space operators, the following integral representation formulas hold: (1) f(A)X - Xf(B)= σ(A)σ(B) f(z) - f(ω)/z - ω E(dz) (AX - XB)F(dw), and Πf(A)X - Xf(B)Π22 (2) = σ(A)σ(B) |f(z) - f(ω)/z - ω|2 ΠE(dz)(AX - XB)F(dw)Π22, 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. © 1999 American Mathematical Society.