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 . 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 and commute are completely characterized. They are shown to be special types of simple weighted shifts. Next, operators of commutator rank one for which 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