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
MathSciNet review: 552265
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 Jselfadjoint 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.