A spectral theorem for -nonnegative operators

Author:
Bernard N. Harvey

Journal:
Trans. Amer. Math. Soc. **257** (1980), 387-396

MSC:
Primary 47B50; Secondary 46D05, 47A45

DOI:
https://doi.org/10.1090/S0002-9947-1980-0552265-5

MathSciNet review:
552265

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Abstract: A *J*-space is a Hilbert space with the usual inner product denoted and an indefinite inner product defined by where *J* is a bounded selfadjoint operator whose square is the identity. We define a *J*-adjoint of an operator *T* with respect to the indefinite inner product in the same way as the regular adjoint is defined with respect to . We say *T* is *J*-selfadjoint if . An operator-valued function is called a *J*-spectral function with critical point zero if it is defined for all , 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 *J*selfadjoint operators *A* with 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 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.

**[1]**M. G. Krein and Ju. L. Smul'jan,*J-polar representation of plus-operators*, Mat. Issled**1**(1966), no. 2, 172-210; English transl., Amer. Math. Soc. Transl. (2)**85**(1969), 115-143. MR**0208373 (34:8183)**

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DOI:
https://doi.org/10.1090/S0002-9947-1980-0552265-5

Article copyright:
© Copyright 1980
American Mathematical Society