\(C\)-semigroups and the abstract Cauchy problem
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 \(X\).
scientific article; zbMATH DE number 150647