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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.

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