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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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##### References:
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