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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Hyperfinite extensions of bounded operators on a separable Hilbert space


Author: L. C. Moore
Journal: Trans. Amer. Math. Soc. 218 (1976), 285-295
MSC: Primary 47A65; Secondary 02H25
DOI: https://doi.org/10.1090/S0002-9947-1976-0402524-0
MathSciNet review: 0402524
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Abstract: Let H be a separable Hilbert space and Ĥ the nonstandard hull of H with respect to an $ {\aleph _1}$-saturated enlargement. Let S be a $ ^\ast$-finite dimensional subspace of $ ^\ast H$ such that the corresponding hyperfinite dimensional subspace Ŝ of Ĥ contains H. If T is a bounded operator on H, then an extension  of T to Ŝ where  is obtained from an internal $ ^\ast$-linear operator on S is called a hyperfinite extension of T. It is shown that T has a compact (selfadjoint) hyperfinite extension if and only if T is compact (selfadjoint). However T has a normal hyperfinite extension if and only if T is subnormal. The spectrum of a hyperfinite extension  equals the point spectrum of Â, and if T is quasitriangular, A can be chosen so that the spectrum of  equals the spectrum of T. A simple proof of the spectral theorem for bounded selfadjoint operators is given using a hyperfinite extension.


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DOI: https://doi.org/10.1090/S0002-9947-1976-0402524-0
Keywords: Nonstandard analysis, Hilbert space, operator, spectrum, subnormal
Article copyright: © Copyright 1976 American Mathematical Society