A spectral theorem for $J$-nonnegative operators

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.