Spectral Theorem for definitizable normal linear operators on Krein spaces

TL;DR

This paper develops a spectral theorem for normal definitizable operators on Krein spaces by constructing a functional calculus analogous to the Hilbert space case.

Contribution

It introduces a functional calculus for normal definitizable operators on Krein spaces, extending spectral theory beyond Hilbert spaces.

Findings

Established a spectral theorem for these operators.

Constructed a functional calculus similar to the Hilbert space case.

Extended spectral analysis techniques to Krein spaces.

Abstract

In the present note a spectral theorem for normal definitizable linear operators on Krein spaces is derived by developing a functional calculus $\phi \mapsto \phi(N)$ which is the proper analogue of $\phi \mapsto \int \phi \, dE$ in the Hilbert space situation.