Strongly commuting selfadjoint operators and commutants of unbounded operator algebras

Schmüdgen, Konrad

doi:10.1090/S0002-9939-1988-0921001-5

Strongly commuting selfadjoint operators and commutants of unbounded operator algebras
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Proc. Amer. Math. Soc. 102 (1988), 365-372 Request permission

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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