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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?)

  • [1] Ralph Philip Boas Jr., Entire functions, Academic Press Inc., New York, 1954. MR 0068627
  • [2] James A. Deddens, Another description of nest algebras, Hilbert space operators (Proc. Conf., Calif. State Univ., Long Beach, Calif., 1977) Lecture Notes in Math., vol. 693, Springer, Berlin, 1978, pp. 77–86. MR 526534
  • [3] Walter Rudin, Real and complex analysis, 2nd ed., McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974. McGraw-Hill Series in Higher Mathematics. MR 0344043

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Article copyright: © Copyright 1980 American Mathematical Society