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

In the present note a spectral theorem for a finite tuple of pairwise commuting, self-adjoint and definitizable bounded linear operators A1,…,An\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_1,\ldots ,A_n$$\end{document} on a Krein space is derived by developing a functional calculus ϕ↦ϕ(A1,…,An)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi \mapsto \phi (A_1,\ldots ,A_n)$$\end{document} which is the proper analogue of ϕ↦∫ϕdE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi \mapsto \int \phi \, dE$$\end{document} in the Hilbert space situation with the common spectral measure E for a finite tuple of pairwise commuting, self-adjoint bounded linear operators.