Functional calculus generated by a square pencil

A linear transformation Υ which assigns to each analytic function defined on the spectrum of the pencil λ ↦ λ2E + λF + H an element of a special Banach algebra is defined. The transformation Υ maps the product of two functions to the product of two elements of the algebra. As an application, as fornula for the solution of the differential equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ E \ddot x(t) + F\dot x(t) + Hx(t) = f(t) $\end{document} is given. Bibliography: 12 titles.