Holomorphic resolvent for integrodifferential equation with completely positive measure (Q1337539)

The title refers to the resolvent (fundamental solution) of the integral equation \[ u(t) = x - \int_{[0,t]} Au (t - s) W(ds), \quad t \geq 0, \] posed in a complex Banach space \(X\). Here \(x \in X\), the operator \(A\) generates a \(C_ 0\)-semigroup on \(X\), and the kernel \(W\) is a scalar completely positive measure, i.e., the inverse \(\psi\) of its Laplace transform satisfies \(\psi (0) = 0\), \(\psi (\infty) = \infty\), and \(\psi'\) is completely monotone. (Thus, \(\psi\) is a Bernstein function, also called an exponent.) The aim is to prove that the resolvent of this equation has a holomorphic extension to a sector containing the positive real axis, and to study the behavior of the resolvent in a neighborhood of the origin. The main existence theorem requires that \(\psi\) asymptotically maps a sector containing the right-half plane into a sector contained in the right-half plane, but it does not require the semigroup generated by \(A\) to be analytic. A converse statement is also proved.