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Proceedings of the American Mathematical Society

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Unitary groups and commutators

Author: Robert M. Kauffman
Journal: Proc. Amer. Math. Soc. 33 (1972), 95-100
MSC: Primary 47.40
MathSciNet review: 0290167
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Abstract: If H is a possibly unbounded selfadjoint operator and A is a closed operator in a Hilbert space, the relation $ (U_t^{ - 1}A{U_t}f)' = iU_t^{ - 1}(AH - HA){U_t}f$ can be shown to hold under relatively reasonable hypotheses on A and f, where $ {U_t} = {e^{iHt}}$. This relation can then be used to relate properties of the commutator $ AH - HA$ to properties of A and H.

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

  • [1] T. Kato, Perturbation theory for linear operators, Die Grundlehren der math. Wissenschaften, Band 132, Springer-Verlag, New York, 1966. MR 34 #3324. MR 0203473 (34:3324)
  • [2] P. Lax and R. S. Phillips, Scattering theory, Pure and Appl. Math., vol. 26, Academic Press, New York, 1967. MR 36 #530. MR 0217440 (36:530)
  • [3] C. R. Putnam, Commutation properties of Hilbert space operators and related topics, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 36, Springer-Verlag, New York, 1967. MR 36 #707. MR 0217618 (36:707)

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Keywords: Unitary group, commutator, positive definite, closed range, finite dimensional null space
Article copyright: © Copyright 1972 American Mathematical Society

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