Strongly commuting selfadjoint operators and commutants of unbounded operator algebras

Author:
Konrad Schmüdgen

Journal:
Proc. Amer. Math. Soc. **102** (1988), 365-372

MSC:
Primary 47D40,; Secondary 47B25,47B47

DOI:
https://doi.org/10.1090/S0002-9939-1988-0921001-5

MathSciNet review:
921001

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Abstract: Let and be (unbounded) selfadjoint operators on a Hilbert space which commute on a dense linear subspace of . To conclude that and strongly commute, additional assumptions are necessary. Two propositions which contain such additional conditions are proved in §1. In §2 we define different commutants of unbounded operator algebras (form commutant, weak unbounded commutant, strong unbounded commutant) and we discuss the relations between them and their bounded parts. In §3 we construct a selfadjoint -representation of the polynomial algebra in two variables for which the form commutant is different from the weak unbounded commutant.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1988-0921001-5

Keywords:
Commutants of unbounded operator algebras,
commuting unbounded selfadjoint operators

Article copyright:
© Copyright 1988
American Mathematical Society