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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 indefiniteinner product. It is shown that these points are stable under compactperturbations. In the second part of the paper we assume that $A$ is symmetricwith respect to the indefinite inner product and prove that the growth of theresolvent of $A$ is of finite order in a neighborhood of a real spectral pointof 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 oftype $\pi_+$ or $\pi_-$.

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