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Proceedings of the American Mathematical Society

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



Hyponormal operators with infinite essential spectrum

Author: Hong W. Kim
Journal: Proc. Amer. Math. Soc. 51 (1975), 44-48
MSC: Primary 47B20
MathSciNet review: 0374974
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Abstract: It is shown that if $ T$ is an essentially hyponormal operator (i.e., the image of $ {T^ \ast }T - T{T^ \ast }$ in the Calkin algebra is a positive element) in $ \mathfrak{L}(\mathcal{H})$, and if the left essential spectrum of $ T$ is infinite, then $ R{({\delta _T})^ - } + \{ {T^ \ast }\} '$ is not norm dense in $ \mathfrak{L}(\mathcal{H})$, where $ R{({\delta _T})^ - }$ denotes the norm closure of the range of derivation induced by $ T$, and $ \{ {T^ \ast }\} '$ denotes the commutant of $ {T^ \ast }$.

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Keywords: Range of a derivation, essential spectrum, norm closure, commutant of an operator, compact operator, hyponormal operator
Article copyright: © Copyright 1975 American Mathematical Society

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