Spectral Points of Type $\pi_+$ and Type $\pi_-$ of Closed Operators in Indefinite Inner Product Spaces

Abstract

We introduce the notion of spectral points of type $\pi_+$ and type $\pi_-$ of closed operators $A$ in a Hilbert space which is equipped with an indefinite inner product. It is shown that these points are stable under compact perturbations. In the second part of the paper we assume that $A$ is symmetric with respect to the indefinite inner product and prove that the growth of the resolvent of $A$ is of finite order in a neighborhood of a real spectral point of type $\pi_+$ or $\pi_-$ which is not in the interior of the spectrum of $A$. Finally, we prove that there exists a local spectral function on intervals of type $\pi_+$ or $\pi_-$.