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 | References | Similar Articles | Additional Information

Abstract: Let *H* be a separable Hilbert space and *Ĥ* the nonstandard hull of *H* with respect to an -saturated enlargement. Let *S* be a -finite dimensional subspace of such that the corresponding hyperfinite dimensional subspace *Ŝ* of *Ĥ* contains *H*. If *T* is a bounded operator on *H*, then an extension *Â* of *T* to *Ŝ* where *Â* is obtained from an internal -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 *Â* equals the point spectrum of *Â*, and if *T* is quasitriangular, *A* can be chosen so that the spectrum of *Â* 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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Additional Information

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