Joint Functional Calculus for Definitizable Self-adjoint Operators on Krein Spaces

Abstract

In the present note a spectral theorem for a finite tuple of pairwise commuting, self-adjoint and definitizable bounded linear operators $$A_1,\ldots ,A_n$$ A 1 , … , A n on a Krein space is derived by developing a functional calculus $$\phi \mapsto \phi (A_1,\ldots ,A_n)$$ ϕ ↦ ϕ ( A 1 , … , A n ) which is the proper analogue of $$\phi \mapsto \int \phi \, dE$$ ϕ ↦ ∫ ϕ d E in the Hilbert space situation with the common spectral measure E for a finite tuple of pairwise commuting, self-adjoint bounded linear operators.