Definitizability of normal operators on Krein spaces and their functional calculus

TL;DR

This paper introduces a new concept of definitizability for normal operators on Krein spaces and develops a corresponding functional calculus, extending classical Hilbert space methods to indefinite inner product spaces.

Contribution

It proposes a novel definitizability concept for normal operators on Krein spaces and constructs an analogous functional calculus to that in Hilbert spaces.

Findings

Established a new definitizability framework for normal operators on Krein spaces.

Developed a functional calculus for these operators similar to the spectral integral in Hilbert spaces.

Extended the theory of operator functional calculus to indefinite inner product spaces.

Abstract

We discuss a new concept of definitizability of a normal operator on Krein spaces. For this new concept we develop a functional calculus $\phi \mapsto \phi(N)$ which is the proper analogue of $\phi \mapsto \int \phi \, dE$ in the Hilbert space situation.