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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Formal Taylor series and complementary invariant subspaces


Author: Domingo A. Herrero
Journal: Proc. Amer. Math. Soc. 45 (1974), 83-87
MSC: Primary 47A60
DOI: https://doi.org/10.1090/S0002-9939-1974-0372662-3
MathSciNet review: 0372662
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Abstract: A class of operators (which includes the unilateral weighted shifts and the noninvertible bilateral weighted shifts on Hilbert spaces), with the property that every element of the commutant of the operator is canonically associated to a formal Taylor series, is characterized. Let $ T$ be one of such operators, then the following result is true: $ T$ has no nontrivial complementary invariant subspaces, no roots and no logarithm.

This result can be partially extended to the case when ``commutant'' is replaced by ``double commutant", then: $ T$ has no nontrivial complementary hyperinvariant subspaces.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1974-0372662-3
Keywords: Formal Taylor series, commutant, double commutant, invariant, bi-invariant, hyperinvariant, complementary invariant subspaces, bilateral and unilateral weighted shifts
Article copyright: © Copyright 1974 American Mathematical Society