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


Diagonalizable normal operators

Author: J. P. Williams
Journal: Proc. Amer. Math. Soc. 54 (1976), 106-108
MathSciNet review: 0397467
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Abstract: If the image $ \varphi (A)$ of a normal operator $ A$ on a separable Hilbert space $ \mathcal{K}$ is a diagonal operator for some nonzero representation $ \varphi $ of $ B(\mathcal{K})$ (that annihilates the compact operators), then $ A$ must itself be a diagonal operator on $ \mathcal{K}$ (with countable spectrum). This yields an ``algebraic'' characterization of the closure of the range of a derivation induced by a diagonal operator.

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Additional Information

PII: S 0002-9939(1976)0397467-0
Keywords: Representations of $ {C^{\ast}}$-algebras, normal operators, range of a derivation
Article copyright: © Copyright 1976 American Mathematical Society