Linear evolution equations in two Banach spaces

Abstract

SynopsisWe consider the ordinary differential equation [Bu(t)]′ = Au(t) with A and B linear operators with domains in a Banach space X and ranges in a Banach space Y. The initial condition is that the limit as (t → 0 of Bu(t) is prescribed in Y. We study the properties of the “solution operator” S(t) which maps the initial state in Y to the solution u(t) at time t. The notion of infinitesimal generator A of S(t) is introduced and the relationships between S(t), an associated semi-group E(t), the operators A and B and some other operators are studied. In particular a pair of operators Ao and Bo, derived from A and B, determine the family S(t) of operators. These so-called “generating pairs” are characterized. The operators A and B and Ao and Bo need not be closed, but form so-called closed pairs which is a weaker condition. We also discuss two applications of the theory.