Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
MathSciNet review: 921001
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] H. Araki and J. P. Jurzak, On a certain class of $ {*}$-algebras of unbounded operators, Publ. Res. Inst. Math. Sci. 18 (1982), 1013-1044. MR 688942 (84g:47041)
  • [2] H. J. Borchers and J. Yngavson, On the algebra of field operators. The weak commutant and integral decomposition of states, Comm. Math. Phys. 42 (1975), 231-252. MR 0377550 (51:13721)
  • [3] P. E. T. Jørgensen and R. T. Moore, Operator commutation relations, Reidel, Dordrecht, 1984.
  • [4] E. Nelson, Analytic vectors, Ann. of Math. (2) 70 (1959), 572-615. MR 0107176 (21:5901)
  • [5] N. S. Poulsen, On the canonical commutation relations, Math. Scand. 32 (1973), 112-122. MR 0327214 (48:5556)
  • [6] -, On $ {C^\infty }$-vectors and intertwining bilinear forms for representations of Lie groups, J. Funct. Anal. 9 (1972), 87-120. MR 0310137 (46:9239)
  • [7] R. T. Powers, Self-adjoint algebras of unbounded operators, Comm. Math. Phys. 21 (1971), 85-124. MR 0283580 (44:811)
  • [8] K. Schmüdgen and J. Friedrich, On commuting unbounded self-adjoint operators. II, J. Integral Equations and Operator Theory 7 (1984), 815-867. MR 774726 (86i:47032)
  • [9] K. Schmüdgen, On commuting unbounded self-adjoint operators. IV, Math. Nachr. 125 (1986), 83 102. MR 847352 (88j:47026)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47D40,, 47B25,47B47

Retrieve articles in all journals with MSC: 47D40,, 47B25,47B47

Additional Information

Keywords: Commutants of unbounded operator algebras, commuting unbounded selfadjoint operators
Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society