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Integral representation formula for generalized normal derivations

Author(s): Danko R. Jocic
Journal: Proc. Amer. Math. Soc. 127 (1999), 2303-2314.
MSC (1991): Primary 47A13, 47B10, 47B15, 47B47, 47B49; Secondary 47A30, 47A60
Posted: April 8, 1999
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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), \label{derF} \end{equation}

and

 \begin{eqnarray}\lefteqn{\|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, \label{derN} \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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Additional Information:

Danko R. Jocic
Affiliation: University of Belgrade, Faculty of Mathematics, Studentski trg 16, P. O. Box 550, 11000 Belgrade, Yugoslavia
Email: jocic@matf.bg.ac.yu

DOI: 10.1090/S0002-9939-99-04802-9
PII: S 0002-9939(99)04802-9
Keywords: Double operator integrals, unitarily invariant norms, Ky-Fan dominance property
Received by editor(s): January 2, 1997
Received by editor(s) in revised form: October 28, 1997
Posted: April 8, 1999
Communicated by: Palle E. T. Jorgensen
Copyright of article: Copyright 1999, American Mathematical Society

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