A commutativity theorem for semibounded operators in hilbert space

Author:
A. Edward Nussbaum

Journal:
Proc. Amer. Math. Soc. **125** (1997), 3541-3545

MSC (1991):
Primary 47B25

DOI:
https://doi.org/10.1090/S0002-9939-97-03977-4

MathSciNet review:
1402881

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Abstract | References | Similar Articles | Additional Information

Abstract: Let and be semibounded (bounded from below) operators in a Hilbert space and a dense linear manifold contained in the domains of , , , and , and such that for all in . It is shown that if the restriction of to is essentially self-adjoint, then and are essentially self-adjoint and and commute, i.e. their spectral projections permute.

**1.**B. Fuglede,*Conditions for two self-adjoint operators to commute or to satisfy the Weyl relation*, Math. Scan.**51**(1982), 163-178. MR**84a:81013****2.**T. Kato,*Perturbation Theory for Linear Operators*, Springer-Verlag, New York, 1966. MR**34:3324****3.**E. Nelson,*Analytic Vectors*, Annals of Mathematics**70**(1959). MR**21:5901****4.**M. Reed and B. Simon,*Functional Analysis in Methods of Modern Mathematical Physics*I, Academic Press, New York and London, 1972. MR**58:12429a****5.**A. E. Nussbaum,*A commutativity theorem for unbounded operators in Hilbert space*, Trans. Amer. Math. Soc.**140**(1969), 485-491. MR**39:3345**

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Additional Information

**A. Edward Nussbaum**

Affiliation:
Department of Mathematics, Washington University, St. Louis, Missouri 63130

Email:
addi@math.wustl.edu

DOI:
https://doi.org/10.1090/S0002-9939-97-03977-4

Received by editor(s):
April 30, 1996

Dedicated:
Dedicated to Allen Devinatz

Communicated by:
Palle E. T. Jorgensen

Article copyright:
© Copyright 1997
American Mathematical Society