Sets of essentially unitary operators

Author:
Ridgley Lange

Journal:
Trans. Amer. Math. Soc. **281** (1984), 65-75

MSC:
Primary 47A15; Secondary 47A65, 47B37

DOI:
https://doi.org/10.1090/S0002-9947-1984-0719659-X

MathSciNet review:
719659

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Abstract: Let be the set of essentially unitary operators on a separable Hilbert space ; for , let be the set of operators such that lies in the Schatten -ideal and the spectrum of does not fill the unit disc; and let be the set of operators in of Fredholm index . The author proves that each is closed and path connected, that is dense in and is path connected for each , and that all these sets are invariant under Cayley transform. It is proved that the spectrum is continuous on but not on , while the spectral radius is continuous on . Sufficient conditions that an operator in have a nontrivial hyperinvariant subspace are given, and it is proved that the general hyperinvariant subspace problem can be reduced to that problem for perturbations of the bilateral shift. The product of commuting operators in is , but this result is false in general. Quasisimilarity in is also studied; quasisimilar operators in are unitarily equivalent modulo the ideal of compacts, and this result also holds in if the spectrum is also preserved.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1984-0719659-X

Keywords:
Essentially unitary operator,
compact operator,
spectrum,
essential spectrum,
hyperinvariant subspace,
shift,
Fredholm operator,
quasisimilar

Article copyright:
© Copyright 1984
American Mathematical Society