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

MathSciNet review:
1402881

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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.**Bent Fuglede,*Conditions for two selfadjoint operators to commute or to satisfy the Weyl relation*, Math. Scand.**51**(1982), no. 1, 163–178. MR**681266****2.**Tosio Kato,*Perturbation theory for linear operators*, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR**0203473****3.**Edward Nelson,*Analytic vectors*, Ann. of Math. (2)**70**(1959), 572–615. MR**0107176****4.**Michael Reed and Barry Simon,*Methods of modern mathematical physics. I. Functional analysis*, Academic Press, New York-London, 1972. MR**0493419****5.**A. E. Nussbaum,*A commutativity theorem for unbounded operators in Hilbert space*, Trans. Amer. Math. Soc.**140**(1969), 485–491. MR**0242010**, 10.1090/S0002-9947-1969-0242010-0

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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:
http://dx.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