Wiener–Hopf type operators and their generalized determinants

Abstract

We recall some results on generalized determinants which support a theory of operator τ-functions in the context of their predeterminants which are operators valued in a Banach-Lie group that are derived from the transition maps of certain Banach bundles. Related to this study is a class of Banach-Lie algebras known as L*-algebras from which several results are obtained in relationship to tau functions. We survey the applicability of this theory to that of Schlesinger systems associated with (operator) equations of Fuschsian type and discuss how meromorphic connections may play a role here.