Operators with small self-commutators

Del Valle, J. W.

doi:10.1090/S0002-9947-1978-0477874-4

Operators with small self-commutators
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by J. W. Del Valle

Trans. Amer. Math. Soc. 240 (1978), 183-194

DOI: https://doi.org/10.1090/S0002-9947-1978-0477874-4

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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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