Second order Banach space valued differential equations: A semigroup approach (Q1423815)

Let \(B\) be the generator of a \(C_0\)-semigroup on a Banach space \(X\). The author gives solutions of the second order equation \(u''(s) = -Bu(s) + f(s)\), where \(f\in C_0({\mathbb R}, X)\) and \(s\in {\mathbb R}\). The main tool is a semigroup technique. Let \(C(t) = T(t)G(t)\) be the product semigroup, where \(T(t)\) is the multiplication semigroup on \(C_0({\mathbb R}, X)\) associated with the semigroup generated by \(B\) and \(G(t)\) is the Gaussian semigroup on \(C_0({\mathbb R}, X)\). The author proves that the spectrum of \(C(t)\) (resp., the spectrum of the generator of \(C(t)\)) is the product of the spectrum of \(T(t)\) and \(G(t)\) (resp., the sum of the spectrum of \(T(t)\) and \(G(t)\)) (Theorem 2.11). This leads to the solutions of the second order problem above.