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\(C\)-semigroups and the abstract Cauchy problem. (English) Zbl 0812.47044
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 \(X\).

MSC:
47D06 One-parameter semigroups and linear evolution equations
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[1] Arendt, W, One parameter semigroups of positive operators, (), 26-47
[2] Davies, E.B; Pang, M.M.H, The Cauchy problem and a generalization of the Hille-Yosida theorem, (), 181-208 · Zbl 0651.47026
[3] \scR. deLaubenfels, C-semigroups and the Cauchy problem, J. Funct. Anal., in press.
[4] deLaubenfels, R, Integrated semigroups, C-semigroups and the abstract Cauchy problem, (), 83-95 · Zbl 0717.47014
[5] Goldstein, J.A, Semigroups of linear operators and applications, (1985), Oxford Univ. Press New York · Zbl 0592.47034
[6] Hille, E; Phillips, R.S, Functional analysis and semigroups, ()
[7] Miyadera, I, On the generators of exponentially bounded C-semigroups, (), 239-242 · Zbl 0617.47032
[8] Neubrander, F, Integrated semigroups and their applications to the abstract Cauchy problem, Pacific J. math., 135, 111-155, (1988) · Zbl 0675.47030
[9] Pazy, A, Semigroups of linear operators and applications to partial differential equations, (1983), Springer-Verlag New York · Zbl 0516.47023
[10] Sanekata, N, Some remarks on the abstract Cauchy problem, Publ. res. inst. math. sci., 11, 51-65, (1975) · Zbl 0328.34065
[11] Tanaka, N, On the exponentially bounded C-semigroups, Tokyo J. math., 10, 107-117, (1987) · Zbl 0631.47029
[12] Tanaka, N; Miyadera, I, Some remarks on C-semigroups and integrated semigroups, (), 139-142 · Zbl 0642.47034
[13] Tanaka, N; Miyadera, I, Exponentially bounded C-semigroups and integrated semigroups, Tokyo J. math., 12, 99-115, (1989) · Zbl 0702.47028
[14] Yosida, K, Functional analysis, (1978), Springer-Verlag New York · Zbl 0217.16001
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