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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

The Fuglede commutativity theorem modulo the Hilbert-Schmidt class and generating functions for matrix operators. I


Author: Gary Weiss
Journal: Trans. Amer. Math. Soc. 246 (1978), 193-209
MSC: Primary 47B15; Secondary 47B10
MathSciNet review: 515536
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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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DOI: http://dx.doi.org/10.1090/S0002-9947-1978-0515536-5
PII: S 0002-9947(1978)0515536-5
Keywords: Fuglede's Theorem, commutators, normal operators, perturbations, Hilbert Schmidt operators, trace class, trace, generating functions for matrices, Laurent matrices, normal derivations
Article copyright: © Copyright 1978 American Mathematical Society