Perturbation theory for dual semigroups. I. The sun-reflexive case
This paper is motivated by the attempt to generalize the ``variation of constant method'' \[ (*)\quad S(t)x=T(t)x+\int^{t}_{0}T(t-s)BS(s)xds. \] Let (A,D(A)) denote the infinitesimal generator of the strongly continuous semigroup \({\mathfrak T}=(T(t))\) of bounded linear operators T(t) on the Banach space X. Then \(X^{\odot}=\overline{D(A')}^{\|.\|}\) is the maximal closed subspace of the dual space X' on which the adjoint semigroup \({\mathfrak T}'\) is strongly continuous. It is denoted by \({\mathfrak T}^{\odot}\). Repeating this construction one gets \({\mathfrak T}^{\odot \odot}\) on \(X^{\odot \odot}\) the latter space containing X as a canonical closed subspace. (X,A) is called sun-reflexive if \(X^{\odot \odot}=X\). If X is reflexive every pair (X,A) is sun-reflexive. Thus the new notion makes non-trivial sense only in the nonreflexive case. The main result reads as follows: Let (X,A) be sun-reflexive and let B be a bounded linear operator from X into X. Then (*) makes sense and defines a new strongly continuous semigroup (S(t)) on X. The theorem is applied to age dependent dynamics. Let us point out that there is another method of extending the given space X to a larger one, introduced by \textit{R. Nagel} [``Sobolev Spaces and Semigroups'', Semi.-Bericht Tübingen, SS 1983, 1-20] and elaborated by \textit{Thomas Walter} [``Abstrakte Sobolev-Räume und ihre Anwendungen auf die Störungstheorie für Generatoren von \(C_ 0\)-Halbgruppen'', Diss. Tübingen (1986)]. This latter method gives new results of similar character as those of the paper under review but even in the case when X is reflexive.
scientific article; zbMATH DE number 4032445
scientific article