Regularity properties for evolution families governed by non-autonomous forms

Abstract

Evolution families are the non-autonomous counterpart of operator semigroups in the well-posedness theory of non-autonomous evolution equations. This note is devoted to fundamental operator theoretical properties, beginning with norm continuity—which we regard as a fingerprint of analyticity. While no known analogue of analytic semigroups is known in the non-autonomous case, we give a sufficient condition for norm-continuity of evolution families. Furthermore, we develop a theory of compact and trace class evolution families. The abstract results are applied to the Laplace operator with time dependent Robin boundary conditions.