Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-1981-0619994-2
MathSciNet review: 619994
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47B15, 47D25

Retrieve articles in all journals with MSC: 47B15, 47D25


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1981-0619994-2
Keywords: Commutator, normal operator, normal derivation, operator ideal, Fuglede property, Schatten $ p$-norms
Article copyright: © Copyright 1981 American Mathematical Society