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.

Invariance of convex sets: An alternative proof and application to Black-Scholes operator

“…on V ′ and let T t be the analytic semigroup generated by −A(t) on V ′ . In order to dodge the consideration of two parameter semigroup theory as raised in [4] or more general multi-parameter one related to attractors as developed in [13], we will proceed otherwise. So, consider the autonomous evolutionary problem…”

Maximal regularity for non autonomous evolution equations and Black-Scholes operator

“…A counterexample was given recently by Fackler [11] generalizing by this way the results of Kalton and Lancien [15] concerning the fail of maximal regularity on various classes of Banach spaces including L p -spaces with 1 < p = 2 < +∞. For an overview on one parameter semigroups, one may see [6] and for more generalized and recent theory on biparametrized semigroups we refer to [3].…”