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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A spectral theorem for $ J$-nonnegative operators


Author: Bernard N. Harvey
Journal: Trans. Amer. Math. Soc. 257 (1980), 387-396
MSC: Primary 47B50; Secondary 46D05, 47A45
MathSciNet review: 552265
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Abstract: A J-space is a Hilbert space with the usual inner product denoted $ [x,y]$ and an indefinite inner product defined by $ (x,y)\, =\, [Jx,y]$ where J is a bounded selfadjoint operator whose square is the identity. We define a J-adjoint $ {T^ + }$ of an operator T with respect to the indefinite inner product in the same way as the regular adjoint $ T^{\ast}$ is defined with respect to $ [x,y]$. We say T is J-selfadjoint if $ T = {T^ + }$. An operator-valued function is called a J-spectral function with critical point zero if it is defined for all $ t \ne 0$, is bounded, J-selfadjoint and has the properties of a resolution of the identity on its domain.

It has been proved by M. G. Krein and Ju. P. Smul'jan that bounded Jselfadjoint operators A with $ (Ax,x) \geqslant 0$ for all x can be represented as a strongly convergent improper integral of t with respect to a J-spectral function with critical point zero plus a nilpotent of index 2. Further, the product of the nilpotent with the J-spectral function on intervals not containing zero is zero.

The present paper extends this theory to the unbounded case. We show that unbounded J-selfadjoint operators with $ (Ax,x) \geqslant 0$ are a direct sum of an operator of the above mentioned type and the inverse of a bounded operator of the same type whose nilpotent part is zero.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1980-0552265-5
PII: S 0002-9947(1980)0552265-5
Article copyright: © Copyright 1980 American Mathematical Society