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Proceedings of the American Mathematical Society

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

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The structure and asymptotic behavior of polynomially compact operators


Author: Frank Gilfeather
Journal: Proc. Amer. Math. Soc. 25 (1970), 127-134
MSC: Primary 47.40
DOI: https://doi.org/10.1090/S0002-9939-1970-0257791-3
MathSciNet review: 0257791
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Abstract: A. R. Bernstein and A. Robinson proved that every polynomially compact operator in Hilbert space has nontrivial invariant subspaces. This paper gives a structure theorem for these operators. We show that a polynomially compact operator is the finite sum of translates of operators which have the property that a finite power of the operator is compact. Furthermore, the spectrum of polynomially compact operators is completely described. Conditions are given to determine the weak and strong asymptotic behavior of a polynomially compact contraction in Hilbert space.


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Keywords: Polynomially compact operator, asymptotic behavior, structure theorem, determination of spectrum, invariant subspaces, hyponormal operator, normal operator
Article copyright: © Copyright 1970 American Mathematical Society