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The Hille-Yosida theorem for local convoluted semigroups. (English) Zbl 1070.47030
Abstract Cauchy problems of the form $u' =Au, \qquad u(0)=x,$ often may have solutions which are not exponentially bounded or even not defined on the whole half-line $$[0,\infty)$$. In such cases, the powerful tool of the Laplace transform is not available to link the evolution operator and the generator $$A$$. Several, to some extent equivalent, approaches to such problems have been developed, ranging from distribution and ultra-distribution semigroups, through locally integrated semigroups, to local convoluted semigroups. This paper deals with local convoluted semigroups. By definition, a closed linear operator on a Banach space $$X$$ is the generator of a local $$F$$-convoluted semigroup on $$[0,T)$$, if the Cauchy problem $u'(t) =Au(t)+F^{[1]}(t)x, \quad 0<t<T,\quad u(0)=0,$ where $$F^{[1]}(t) =\int_0^tF(s)ds$$, has a unique classical solution for any $$x \in X$$. The family $$(S(t))_{t\geq 0}$$ defined by $$S(t)x =u'(t)$$ is called the local $$F$$-convoluted semigroup generated by $$A$$. This definition yields $$C_0$$-semigroups for the choice $$F^{[1]}(t)=1$$ for all $$t$$, whereas, e.g., distribution semigroups correspond to $$F^{[1]}(t) = t^\alpha/\Gamma(\alpha+1), \alpha\geq 0$$.
The main result of this paper is a Hille-Yosida type theorem characterising generators of local convoluted semigroups. Without going into details, it states that under certain technical assumptions, $$(S(t))_{t\geq 0}$$ is the local $$F$$-convoluted semigroup on $$[0,T)$$ generated by $$A$$ if and only if for some $$\omega\in \mathbb{R}$$, $$(\omega,\infty)\subset \rho(A)$$ and $$(\omega,\infty)\ni \lambda \to \int_0^\tau e^{-\lambda t}dF(t)\cdot R(\lambda,A)$$ is the local Laplace-Stieltjes transform on $$[0,\tau]$$, $$\tau<T$$, of $$(S(t))_{t\geq 0}$$.
The paper is concluded by specializing this result to the exponentially bounded case.

##### MSC:
 47D03 Groups and semigroups of linear operators 47D06 One-parameter semigroups and linear evolution equations 44A10 Laplace transform
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