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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

A note on direct sums of quasinilpotent operators


Author: L. A. Fialkow
Journal: Proc. Amer. Math. Soc. 48 (1975), 125-131
MSC: Primary 47A65
MathSciNet review: 0365201
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Abstract: Let $ \mathcal{Q}$ denote the set of all quasinilpotent operators on a fixed, separable Hilbert space $ \mathcal{H}$. D. Herrero has found necessary conditions for an operator to belong to the norm closure of $ \mathcal{Q}$ in $ \mathcal{L}(\mathcal{H})$. We prove that each direct sum (or direct integral) of operators in $ {\mathcal{Q}^ - }$ satisfies these conditions; two questions of D. Herrero are thereby related to one another. We also prove that the spectrum of a direct sum of nilpotent operators may be multiply connected.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0365201-5
PII: S 0002-9939(1975)0365201-5
Keywords: Quasinilpotent operator, connected spectrum, connected essential spectrum
Article copyright: © Copyright 1975 American Mathematical Society