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Proceedings of the American Mathematical Society

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On a boundedness condition for operators with a singleton spectrum

Author: J. P. Williams
Journal: Proc. Amer. Math. Soc. 78 (1980), 30-32
MSC: Primary 47A30; Secondary 47A65
MathSciNet review: 548078
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Abstract: For a bounded invertible linear operator A let $ {\mathcal{B}_A}$ consist of those operators X for which $ \sup \{ \left\Vert {{A^n}X{A^{ - n}}} \right\Vert:n \geqslant 0\} > \infty $. It is shown that $ {\mathcal{B}_A}$ contains the ideal of compact operators if and only if A is similar to a scalar multiple of a unitary operator. Also, if A is invertible and either has a one-point spectrum or is positive definite then $ {\mathcal{B}_A} \cap {\mathcal{B}_{{A^{ - 1}}}}$ is the commutant of A.

References [Enhancements On Off] (What's this?)

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Article copyright: © Copyright 1980 American Mathematical Society

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