Functional calculus generated by a square pencil

A linear transformation $\Upsilon$ is introduced that assigns an element of a special Banach algebra to each analytic function defined on the spectrum of the pencil $\lambda\mapsto\lambda^2E+\lambda F+H$. The transformation $\Upsilon$ maps the product of two functions into the product of two elements of the algebra. As an application, a formula for a solution of the differential equation $E\ddot x(t)+F\dot x(t)+Hx(t)=f(t)$ is given.