B-evolusies en die laplace transformasie

Abstract

If Bis a linear operator with domain DCB) contained in a Banach space X and range in a Banach space Y, the family {S(t):t>O} of bounded, linear operators defined on Y is called a B-evolution if S(t)CYJ c D(B) and SCt+s) = S<t>BS(s) for all positive t ands. Associated with SCt> is the semi-group {E(t):t>O} of linear operators in Y, defined by ECt) = BS(t). In this dissertation the properties of a-evolutions are studied. Certain assumptions are made with respect to S(t) and E Ct>. The infinitesimal operator Ao of S(t) is de= fined and it is shown that restrictions of Ao and the operator B form a closable pair. The closure of this pair is denoted by <a,e> and it is shown that the operator ~e - a, with Re~>O, maps PC~)[YJ onto Y and where P<~> is the Laplace transform of S<t>. The closed pair <a,e> will determine the B-evolution uniquely only if S<t> is strongly continuous for t>O. The initial states y for which u<t> = S<t>y solves tha Cauchy problem are also determined. DtCBu(t)J = Aou(t) Bu(t>lt-o = y. The link between S<t> and E<t> is studied. An unbounded operator C, that links the B-evolution S(t) and its associated semi-group E<t>, is constructed such that S(t) = CE<t>. Finally, the concept of a family of operators which is in empathy with a semi-group is introduced. Such families are studied, and conditions determined under which they are B-evolutions.