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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The Fuglede commutativity theorem modulo operator ideals

Author: Gary Weiss
Journal: Proc. Amer. Math. Soc. 83 (1981), 113-118
MSC: Primary 47B15; Secondary 47D25
MathSciNet review: 619994
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Abstract: Let $ H$ denote a separable, infinite-dimensional complex Hilbert space. A two-sided ideal $ I$ of operators on $ H$ possesses the generalized Fuglede property (GFP) if, for every normal operator $ N$ and every $ X \in L(H)$, $ NX - XN \in I$ implies $ {N^ * }X - X{N^ * } \in I$. Fuglede's Theorem says that $ I = \left\{ 0 \right\}$ has the GFP. It is known that the class of compact operators and the class of Hilbert-Schmidt operators have the GFP.

We prove that the class of finite rank operators and the Schatten $ p$-classes for $ 0 < p < 1$ fail to have the GFP. The operator we use as an example in the case of the Schatten $ p$-classes is multiplication by $ z + w$ on $ {L^2}$ of the torus.

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Keywords: Commutator, normal operator, normal derivation, operator ideal, Fuglede property, Schatten $ p$-norms
Article copyright: © Copyright 1981 American Mathematical Society

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