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Reflexivity of the commutant and local commutants of an algebraic operator

✍ Scribed by Janko Bračič

Publisher: Elsevier Science
Year: 2007
Tongue: English
Weight: 153 KB
Volume: 420
Category: Article
ISSN: 0024-3795
DOI: 10.1016/j.laa.2006.05.002

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✦ Synopsis

We show that an algebraic operator on a complex Banach space has reflexive commutant if and only if each zero of the minimal polynomial of the operator is simple. Further, for any operator, the local commutant at an eigenvector is reflexive. On the other hand, for an algebraic operator whose minimal polynomial has at least one zero that is not simple, the local commutant of the operator at a given vector is reflexive precisely when the vector is an eigenvector.

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