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Some remarks on C-semigroups and integrated semigroups. (English) Zbl 0642.47034
Let X be a Banach space and let B(X) be the set of all bounded linear operators from X into itself. Let C be an injective operator in B(X) and the range R(C) be dense in X.
A family \(\{\) S(t);t\(\geq 0\}\) in B(X) is called C-semigroup, if
I) \(S(t+s)C=S(t)S(s)\) for t,s\(\geq 0\), \(S(0)=C,\)
II) \(S(\cdot)X:(0,+\infty)\to X\) is continuous for \(x\in X,\)
III) there are \(M\geq 0\) and \(a\in R\) such that \(\| s(t)\| Me^{at}\) for \(t\geq 0.\)
If \(G(x)=\lim_{t\to 0^+}(C^{-1}S(t)x-x)/t\) for \(x\in D(G)\), it is known that G is densely defined and closable, the closure \(\bar G\) is called the C-c.i.g of \(\{\) S(t);t\(\geq 0\}.\)
Let n be a positive integer, a family \(\{\) U(t):t\(\geq 0\}\) in B(X) is called n-times integrated semigroup, if
I) \(U(\cdot)x:\{0,+\infty)\to X\) is continuous for \(x\in X.\)
II) \(U(t)U(s)x=1/(n-1)!(\int^{t+s}_{t}(s+t-r)^{n-1}U(r)xdr- \int^{s}_{0}(s+t-r)^{n-1}U(r)xdr)\) for \(x\in X\) and s,t\(\geq 0\) and \(U(0)=0,\)
III) \(U(t)x=0\), for all \(t>0\), implies \(x=0,\)
IV) there are \(M\geq 0\) and \(\omega\in R\) such that \(\| U(t)\| \leq Me^{\omega t}\) for \(t\geq 0.\)
A very good result about the relation between these two kinds of semigroups is the following:
Theorem. Let A be a densely defined closed linear operator in X with \(\rho\) (A)\(\neq \emptyset\). Let \(c\in \rho (A)\) and \(n\geq 0\) be an integer. The following (i)-(iii) are equivalent:
i) A is the generator of an n-times integrated semigroup \(\{\) U(t);t\(\geq 0\},\)
ii) A is the C-c.i.g of a C-semigroup \(\{\) S(t);t\(\geq 0\}\) with \(C=R(C;A)^ n,\)
iii) There exist \(M\geq 0\) and \(a\in R\) such that (a,\(\infty)\subset \rho (A)\) and \(\| R((\lambda;A)^ mR(c;A)^ n\| \leq M/(\lambda -a)^ m\) for \(m\geq 1\) and \(\lambda >a.\)
In this case, we have \[ U(t)x=(c-A)^ n\int^{t}_{0}\int^{t_ 1}_{0}...\int^{t_{n-1}}_{0}s(t_ n)xdt_ n...dt_ 2dt_ 1 \] for \(x\in X\) and \(t\geq 0\).
Reviewer: Wu Liangsen

MSC:
47D03 Groups and semigroups of linear operators
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