On the exponentially bounded C-semigroups. (English) Zbl 0631.47029

Tanaka, Naoki

Tokyo J. Math. 10, No. 1-2, 107-117 (1987).

If C is an injective bounded operator with dense range on a Banach space, one defines an exponentially bounded C-semigroup as a strongly continuous family of bounded operators, S(t), \(t\geq 0\), such that \(S(t+s)C=S(t)S(s)\), \(S(0)=C\), and \(\| S(t)\| \leq Me^{at}\) [E. B. Davies and M. M. N. Pang, The Cauchy problem and a generalization of the Hille-Yosida theorem (to appear)]. The C-complete infinitesimal generator (C-c.i.g.) of S(t) is the closure \(\bar G\) of the “derivative” at \(t=0\) of \(C^{-1}S(t)\). Representation theorems of S(t) in terms of \(\bar G\) are proved. Also, necessary and sufficient conditions (generalizing those for \((C_ 0)\)-semigroups) for a closed operator to be a C-c.i.g. are given. It is shown that, if the abstract Cauchy problem: \(u'(t)=Au(t)\), \(u(0)=x\), where A and C commute, has a unique solution with \(\| u(t)\| \leq Me^{at}\| C^{-1}x\|\) for all \(x\in CD(A)\), and if CD(A) is a core of A, then A is a C-c.i.g. (The converse of a result by Davies-Pang.) Finally, connections with semigroups of growth order \(\alpha\) are established.

Reviewer: N.Angelescu

MSC:

47D03

Groups and semigroups of linear operators

Keywords:

injective bounded operator with dense range on a Banach space; exponentially bounded C-semigroup; C-complete infinitesimal generator; abstract Cauchy problem; semigroups of growth order \(\alpha \)