\(B\)-bounded semigroups and implicit evolution equations (Q5936448)

\(B\)-bounded semigroups \((Y(t))\) of linear operators generated by \(A\) on a Banach space \(X\) are closely related to an abstract Cauchy problem defined by \((A,D(A))\) on \(X\) and a second linear operator \((B,D(B))\) from \(X\) to a further Banach space \(Z\), fulfilling an evolution equation NEWLINE\[NEWLINEY(t)f= Bf+ \int^t_0 Y(s) Af dsNEWLINE\]NEWLINE for \(f\) in a subspace of \(D(A)\cap D(B)\). [For details see e.g. \textit{A. Belleni-Morante}, Ann. Mat. Pura Appl., IV. Ser. 170, 359-376 (1996; Zbl 0882.47012)] and the author's previous investigation [ibid. 175, 307-326 (1998; Zbl 1005.47042)]. It is not assumed that \(A\) generates a semigroup \((\exp(tA))_{t\geq 0}\), but under additional assumptions the restrictions to suitable intermediate subspaces fulfil resolvent conditions similar to the Hille-Yoshida theory.NEWLINENEWLINENEWLINEIf \(B\) is injective, closable with bounded inverse the description of the generator of \(B\)-bounded semigroups is simplified (Theorem 2.11). The role of \(B\)-bounded semigroups for solutions of abstract Cauchy problems is described in Section 3. E.g., if \(X\) is embedded into a further Banach space \(\widetilde X\), and \(\widetilde K\), \(\widetilde L\) are linear operators from \(Z\) to \(\widetilde X\), then an \(\widetilde X\) solution of the corresponding Cauchy problem is a solution of \({d\over dt}(\widetilde Ku)= \widetilde Lu\) with the initial condition \(\widetilde Ku(0+)= u^0\) (in the topology of \(\widetilde X\)).NEWLINENEWLINENEWLINEIt is shown (in Theorem 3.2) that -- under additional conditions -- \(B\)-bounded semigroups generated by \(A\) are representable as \(\widetilde X\) solutions of an abstract Cauchy problem for suitable auxiliary spaces \(\widetilde X= X_B\).NEWLINENEWLINENEWLINEThe discussion of this result, and connections to other approaches, e.g. empathy theory [cf. \textit{N. Sauer}, Banach Center Publ., 38, Warsaw, 325-338 (1997; Zbl 0885.47013)] are the subjects investigated in Section 3.