On the exponentially bounded C-semigroups
type: Item
instance of: scholarly article
title: On the exponentially bounded C-semigroups
zbMATH Open document ID: 0631.47029
author: Naoki Tanaka
published in: Tokyo Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: 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 \(\| S(t)\| \leq Me^{at}\) [\textit{E. B. Davies} and \textit{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.
Mathematics Subject Classification ID: 47D03
zbMATH DE Number: 4026186
zbMATH Keywords: injective bounded operator with dense range on a Banach space
zbMATH Keywords: exponentially bounded C-semigroup
zbMATH Keywords: C-complete infinitesimal generator
zbMATH Keywords: abstract Cauchy problem
zbMATH Keywords: semigroups of growth order \(\alpha \)
reviewed by: Nicolae Angelescu
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270141795
OpenAlex ID: W1987068669
DOI: 10.3836/TJM/1270141795
description: scientific article; zbMATH DE number 4026186