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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Operators with small self-commutators
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by J. W. Del Valle PDF
Trans. Amer. Math. Soc. 240 (1978), 183-194 Request permission

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
  • © Copyright 1978 American Mathematical Society
  • 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