# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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## The Fuglede commutativity theorem modulo the Hilbert-Schmidt class and generating functions for matrix operators. IHTML articles powered by AMS MathViewer

by Gary Weiss
Trans. Amer. Math. Soc. 246 (1978), 193-209 Request permission

## Abstract:

We prove the following statements about bounded linear operators on a separable, complex Hilbert space: (1) Every normal operator N that is similar to a Hilbert-Schmidt perturbation of a diagonal operator D is unitarily equivalent to a Hilbert-Schmidt perturbation of D; (2) For every normal operator N, diagonal operator D and bounded operator X, the Hilbert-Schmidt norms (finite or infinite) of $NX - XD$ and ${N^{\ast }}X - X{D^{\ast }}$ are equal; (3) If $NX - XN$ and ${N^{\ast }}X - X{N^{\ast }}$ are Hilbert-Schmidt operators, then their Hilbert-Schmidt norms are equal; (4) If X is a Hilbert-Schmidt operator and N is a normal operator so that $NX - XN$ is a trace class operator, then Trace$\left ( {NX - XN} \right ) = 0$; (5) For every normal operator N that is a Hilbert-Schmidt perturbation of a diagonal operator, and every bounded operator X, the Hilbert-Schmidt norms (finite or infinite) of $NX - XN$ and ${N^{\ast }}X - X{N^{\ast }}$ are equal. The main technique employs the use of a new concept which we call ’generating functions for matrices’.
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