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Intersections of commutants with closures of derivation ranges


Author: Domingo A. Herrero
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
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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

DOI: https://doi.org/10.1090/S0002-9939-1979-0521868-3
Keywords: Inner derivations, norm closure, weak closure, $ {\text{weak}^\ast}$ closure, closure of the range of an inner derivation, commutant, normal eigenvalue, biquasitriangular operators
Article copyright: © Copyright 1979 American Mathematical Society

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