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Proceedings of the American Mathematical Society

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



Finite operators

Author: J. P. Williams
Journal: Proc. Amer. Math. Soc. 26 (1970), 129-136
MSC: Primary 47.40
MathSciNet review: 0264445
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Abstract: A bounded linear operator $ A$ on a Hilbert space $ H$ is called finite if $ \vert\vert AX - XA - 1\vert\vert \geqq 1$ for each $ X \in B(H)$. The class of finite operators is uniformly closed, contains every normal operator, every operator with a compact direct summand, and the entire $ {C^ \ast }$-algebra generated by each of its members. These results imply that the set of operators with a finite dimensional reducing subspace is not uniformly dense. It is also shown that the set of self-commutators is uniformly closed.

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Keywords: Commutators, reducible operators, numerical range, $ {C^ \ast }$-algebras
Article copyright: © Copyright 1970 American Mathematical Society