A theoretical framework for two-parameter semigroups

Abstract

Here, we present a theoretical framework for two-parameter semigroups of bounded linear operators on a Banach space. Our approach relies on a new definition of the infinitesimal generator of two-parameter semigroups. This definition, in the case of C0-two parameter semigroups, allows trajectory to be differentiable on the nonnegative cone of the plane, when the initial state is in the domain of this generator. We provide also the abstract Cauchy problem satisfied by these trajectories. We prove some theoretical and general results concerning relationships between this generator and the infinitesimal generators of the components. We investigate commutativity relations and precise the domains of their validity. We establish the extension of the Hille-Yoshida known for one-parameter semigroups. We provide some examples and we give an application to the product semigroup.