Functional calculus generated by a square pencil (Q690587)

Let \(E,F,H\) be bounded linear operators acting from a Banach space \(X\) to a Banach space \(Y\). The function \(\lambda\rightarrow \lambda^2E+\lambda F+\lambda H\) is called a square pencil. The spectrum of a square pencil is the set \(\sigma( E, F, H)\) of all complex \(\lambda\) for which the operator \(\lambda^2E+\lambda F+\lambda H\) has no inverse. The author constructs a functional calculus \(\Upsilon\) from the space of functions which are analytic on the spectrum of a quadratic pencil, into the Banach algebra \(B_{(E,F,H)}(X,Y)\). By applying the functional calculus, a solution for the differential equation \( E \ddot x(t) + F\dot x(t) + Hx(t) = f(t) \) is obtained.