On a boundedness condition for operators with a singleton spectrum
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- by J. P. Williams PDF
- Proc. Amer. Math. Soc. 78 (1980), 30-32 Request permission
Abstract:
For a bounded invertible linear operator A let ${\mathcal {B}_A}$ consist of those operators X for which $\sup \{ \left \| {{A^n}X{A^{ - n}}} \right \|:n \geqslant 0\} > \infty$. 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
- Ralph Philip Boas Jr., Entire functions, Academic Press, Inc., New York, 1954. MR 0068627
- 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
- Walter Rudin, Real and complex analysis, 2nd ed., McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974. MR 0344043
Additional Information
- © Copyright 1980 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 78 (1980), 30-32
- MSC: Primary 47A30; Secondary 47A65
- DOI: https://doi.org/10.1090/S0002-9939-1980-0548078-6
- MathSciNet review: 548078