On subordinated holomorphic semigroups.

*(English)*Zbl 0743.47017The 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.

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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\textit{A. S. Carasso} and \textit{T. Kato}, Trans. Am. Math. Soc. 327, No. 2, 867--878 (1991; Zbl 0743.47017)

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