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Proceedings of the American Mathematical Society

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

 

 

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 $ {A_1}$ and $ {A_2}$ be (unbounded) selfadjoint operators on a Hilbert space $ \mathcal{H}$ which commute on a dense linear subspace of $ \mathcal{H}$. To conclude that $ {A_1}$ and $ {A_2}$ 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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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