Pseudodifferential operators with non-regular operator-valued symbols

In this paper, we consider pseudodifferential operators with operator-valued symbols and their mapping properties, without assumptions on the underlying Banach space E. We show that, under suitable parabolicity assumptions, the Wpk(Rn,E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${W_p^k(\mathbb{R}^n, E)}$$\end{document}-realization of the operator generates an analytic semigroup. Our approach is based on oscillatory integrals and kernel estimates for them. An application to non-autonomous pseudodifferential Cauchy problems gives the existence and uniqueness of a classical solution. As an example, we include a discussion of coagulation–fragmentation processes.