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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Intersections of commutants with closures of derivation ranges
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by Domingo A. Herrero PDF
Proc. Amer. Math. Soc. 74 (1979), 29-34 Request permission

Abstract:

The norm closure of the set ${\mathcal {A}_w}(\mathcal {X}) = \cup \;\{ {\text {Ran}}{({\delta _A})^{ - w}} \cap \{ A\} ’:A \in \mathcal {L}(\mathcal {X})\}$, where ${\delta _A}$ denotes the inner derivation induced by the operator A, ${\text {Ran}}{({\delta _A})^{ - w}}$ is the weak closure of the range of ${\delta _A}$ and $\{ A\} ’$ is the commutant of A, is disjoint from the open dense subset $\mathcal {B}(\mathcal {X}) = \{ T \in \mathcal {L}(\mathcal {X})$: T has a nonzero normal eigenvalue} for every complex Banach space $\mathcal {X}$. For a Hilbert space $\mathcal {H}$, $\mathcal {L}(\mathcal {H}) = \mathcal {B}(\mathcal {H}) \cup {\mathcal {A}_w}{(\mathcal {H})^ - }$, where the bar denotes norm closure.
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Additional Information
  • © Copyright 1979 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 74 (1979), 29-34
  • MSC: Primary 47B47
  • DOI: https://doi.org/10.1090/S0002-9939-1979-0521868-3
  • MathSciNet review: 521868