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Transactions of the American Mathematical Society

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Operators with small self-commutators


Author: J. W. Del Valle
Journal: Trans. Amer. Math. Soc. 240 (1978), 183-194
MSC: Primary 47B47; Secondary 47A65
DOI: https://doi.org/10.1090/S0002-9947-1978-0477874-4
MathSciNet review: 0477874
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Abstract: Let A be a bounded operator on a Hilbert space H. The self-commutator of A, denoted [A], is $ {A^\ast}A - A{A^\ast}$. An operator is of commutator rank n if the rank of [A] is n. In this paper operators of commutator rank one are studied. Two particular subclasses are investigated in detail. First, completely nonnormal operators of commutator rank one for which $ {A^\ast}A$ and $ A{A^\ast}$ commute are completely characterized. They are shown to be special types of simple weighted shifts. Next, operators of commutator rank one for which $ \{ {A^n}e\} _{n = 0}^\infty $ is an orthogonal sequence (where e is a generator of the range of [A]) are characterized as a type of weighted operator shift.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1978-0477874-4
Keywords: Operator, normal, hyponormal, commutator
Article copyright: © Copyright 1978 American Mathematical Society

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