Integral representation formula for generalized normal derivations

Jocić, Danko

doi:10.1090/S0002-9939-99-04802-9

Integral representation formula for generalized normal derivations
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by Danko R. Jocić PDF

Proc. Amer. Math. Soc. 127 (1999), 2303-2314 Request permission

Abstract:

For generalized normal derivations, acting on the space of all bounded Hilbert space operators, the following integral representation formulas hold: \begin{equation} f(A)X-Xf(B)=\int _{ \sigma (A)}\int _{\sigma (B)}\frac {f(z)-f(w)}{z-w}\,E(dz)\,(AX-XB)F(dw), \end{equation} and \begin{eqnarray} {\|f(A)X-Xf(B)\|_2^2}\nonumber \ & & =\int _{ \sigma (A)}\int _{\sigma (B)}\left \vert \frac {f(z)-f(w)}{z-w}\right \vert ^2 \,\|E(dz)(AX-XB)F(dw)\|_2^2, \end{eqnarray} whenever $AX-XB$ is a Hilbert-Schmidt class operator and $f$ is a Lipschitz class function on $\sigma (A)\cup \sigma (B).$ 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.

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