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Proceedings of the American Mathematical Society

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

 

 

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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DOI: https://doi.org/10.1090/S0002-9939-1970-0257791-3
Keywords: Polynomially compact operator, asymptotic behavior, structure theorem, determination of spectrum, invariant subspaces, hyponormal operator, normal operator
Article copyright: © Copyright 1970 American Mathematical Society