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On subordinated holomorphic semigroups. (English) Zbl 0743.47017
The paper studies semigroups of probability measures that are holomorphic. Let $$X$$ be a complex Banach space and $$C_ 0(X)$$ the class of uniformly bounded $$C_ 0$$-semigroups $$[T(t)]$$, $$t\geq 0$$ on $$X$$. Let $$B(X)$$ be the Banach algebra of bounded linear operators on $$X$$, let $$S$$ be the Banach algebra of complex Borel measures $$\mu$$ on $$\mathbb{R}^ +$$, with convolution and multiplication, and normed by the total variation. If $$V$$ is a Borel set $$\subset\mathbb{R}^ +$$, $$\mu(V)$$ denotes the value of $$\mu$$ on $$V$$, whereas $$\int_ V g(u)\mu(du)$$ is the integral with respect to $$\mu$$ of the Borel measurable function $$g$$. Let $$L$$ be the Banach space of Borel measurable functions $$f$$ on $$\mathbb{R}^ +$$. For each $$\mu\in S$$, define $$Z_ \mu\in B(L)$$ by $$Z_ \mu f=(\mu*f)(\tau)=\int_{\mathbb{R}^ +} f(\tau-u)\mu(du)$$, $$f\in L$$, $$\tau\geq 0$$. For each $$\mu\in S$$ define $$\langle\mu,T\rangle=\int_{\mathbb{R}^ +} T(u)\mu(du)$$. Then, $$\mu\to\langle\mu,T\rangle$$ is a continuous homomorphism of $$S$$ into $$B(X)$$.
Let $$P$$ be the set of all algebraic semigroups $$[p(t)]$$, $$t\geq 0$$ of probability measures on $$\mathbb{R}^ +$$. If $$[p(t)]\in P$$, then $$[Z_ p(t)]=[Z_{p(t)}]$$ forms an algebraic contraction semigroup on $$L$$, and for $$[T(t)]\in C_ 0(X)$$, $$[U(t)]=[\langle p(t),T\rangle]$$ is a uniformly bounded algebraic semigroup on $$X$$. $$[U(t)]$$ is said subordinated to $$[T(t)]$$.
The paper shows a large class of semigroups $$[p(t)]$$, $$t\geq 0$$ of probability measures such that $$[U(t)]$$ is holomorphic whenever $$T(t)\in C_ 0(X)$$ and constructs families $$[p(t)]$$ that do not have this property.
Reviewer: S.Totaro (Firenze)

##### MSC:
 47D03 Groups and semigroups of linear operators 60J25 Continuous-time Markov processes on general state spaces 60E07 Infinitely divisible distributions; stable distributions
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