Functional Calculus for Definitizable Self-adjoint Linear Relations on Krein Spaces

Abstract

In the present note a functional calculus $${\phi \mapsto \phi(A)}$$ for self-adjoint definitizable linear relations on Krein spaces is developed. This functional calculus is the proper analogue of $${\phi \mapsto \int \phi \, dE}$$ in the Hilbert space situation where $${\phi}$$ is a bounded and measurable function on $${\sigma(A)}$$ and $${\int \phi \, dE}$$ is defined in the weak sense. The derived functional calculus also comprises the Spectral Theorem for self-adjoint definitizable operators on Krein spaces showing the existence of spectral projections.