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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by Konrad Schmüdgen

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

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

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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.

References

Huzihiro Araki and Jean-Paul Jurzak, On a certain class of $\ast$-algebras of unbounded operators, Publ. Res. Inst. Math. Sci. 18 (1982), no. 3, 1013–1044. MR 688942, DOI 10.2977/prims/1195183293
H. J. Borchers and J. Yngvason, On the algebra of field operators. The weak commutant and integral decompositions of states, Comm. Math. Phys. 42 (1975), 231–252. MR 377550, DOI 10.1007/BF01608975

Operator commutation relations

Edward Nelson, Analytic vectors, Ann. of Math. (2) 70 (1959), 572–615. MR 107176, DOI 10.2307/1970331
Niels Skovhus Poulsen, On the canonical commutation relations, Math. Scand. 32 (1973), 112–122. MR 327214, DOI 10.7146/math.scand.a-11448
Neils Skovhus Poulsen, On $C^{\infty }$-vectors and intertwining bilinear forms for representations of Lie groups, J. Functional Analysis 9 (1972), 87–120. MR 0310137, DOI 10.1016/0022-1236(72)90016-x
Robert T. Powers, Self-adjoint algebras of unbounded operators, Comm. Math. Phys. 21 (1971), 85–124. MR 283580, DOI 10.1007/BF01646746
K. Schmüdgen and J. Friedrich, On commuting unbounded selfadjoint operators. II, Integral Equations Operator Theory 7 (1984), no. 6, 815–867. MR 774726, DOI 10.1007/BF01195869
Konrad Schmüdgen, On commuting unbounded selfadjoint operators. IV, Math. Nachr. 125 (1986), 83–102. MR 847352, DOI 10.1002/mana.19861250107