\(C\)-semigroups and the abstract Cauchy problem
type: Item
instance of: scholarly article
title: \(C\)-semigroups and the abstract Cauchy problem
zbMATH Open document ID: 0812.47044
DOI: 10.1016/0022-247X(92)90013-4
author: Naoki Tanaka
author: Isao Miyadera
published in: Journal of Mathematical Analysis and Applications
publication date: 1993-04-01T00:00:00Z
review text: Let \(C\) be a bounded injective (linear) operator on a Banach space \(X\). A strongly continuous family \(S= \{S(t): t\geq 0\}\subset B(X)\) is a \(C\)- semigroup if \(S(t+ s) C= S(t) S(s)\), \(S(s)= C\). It is exponentially bounded if also \(\| S(t)\|\leq Me^{at}\) holds. With \(A= C^{-1} S'(0)\), \(u(t)= C^{-1} S(t)y\) gives the unique solution of \(u'(t)= Au(t)\), \(u(0)= y\) with \(y= (\lambda- A)^{-1} Cx\) when \(\lambda- A\) is injective (usually \(\lambda\in \rho(A)\)). The theory of \(C\)-semigroups was originated by G. DaPrato and later independently rediscovered by E. B. Davies and M. Pang. Many authors have contributed to this theory, including W. Arendt, R. de Laubenfels, M. Hieber, F. Neubrander, the authors, and others. This paper surveys the connections between the abstract Cauchy problem and \(C\)-semigroups, and the generation theory of \(C\)-semigroups. The emphasis is on generality, thus for example, \(S\) may not be exponentially bounded, and the range of \(C\) need not be dense in \(X\).
reviewed by: Jerome A. Goldstein
Mathematics Subject Classification ID: 47D06
zbMATH DE Number: 150647
zbMATH Keywords: \(C\)-semigroup
zbMATH Keywords: exponentially bounded
zbMATH Keywords: abstract Cauchy problem
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 150647