Functional Calculus for definitizable self-adjoint linear relations on Krein spaces

TL;DR

This paper develops a functional calculus for self-adjoint definitizable linear relations on Krein spaces, extending spectral theory analogous to Hilbert spaces and establishing spectral projections.

Contribution

It introduces a functional calculus for definitizable relations on Krein spaces, generalizing spectral theory to indefinite inner product spaces.

Findings

Established a functional calculus for self-adjoint definitizable relations

Proved the existence of spectral projections in Krein spaces

Extended spectral theorem to the Krein space setting

Abstract

In the present note a functional calculus $\phi \mapsto \phi(A)$ for self-adjoint definitizable linear relation on Krein spaces is developed. This functional calculus is the proper analogue of $\phi \mapsto \int \phi \, dE$ in the Hilbert space situation. It also comprises the Spectral Theorem for self-adjoint definitizable operators on Krein spaces showing the existence of spectral projections.