Analyticity and injectivity of convolution semigroups on Lie groups
A family \(\{\mu_t\), \(t > 0\}\) of probability distributions on a Lie group \(G\) is called a continuous convolution semigroup if \(\mu_s*\mu_t = \mu_{s+t}\) and \(\mu_t\) converges weakly to \(\delta_e\) as \(t\to 0\). Such a semigroup generates the semigroup of linear operators \(\{T(t)\), \(t> 0\}\) in the space of complex-valued continuous functions on \(G\) given by \[ T(t)f(x) = \int_G f(xy)\mu_t(dy). \] Let \(A\) be its infinitesimal operator. The following two theorems are proved. The semigroup \(T(t)\) is analytic whenever \(A\) is nondegenerate or \(\mu_t,\), \(t> 0\), are symmetric. For any continuous semigroup \(\mu_t\), the semigroup \(T(t)\) is injective, i.e., \(\text{Ker }T(t)= 0\) for all \(t > 0\). This is well known for \(G = \mathbb{R}^n\) and was proved for compact \(G\) by the author [Osaka J. Math. 34, No. 1, 233-264 (1997; Zbl 0876.60003)]. It follows that, for any \(t > 0\), \(\mu_t*\xi = \mu_t*\xi'\) implies \(\xi = \xi'\). As a corollary, for any infinitely divisible distribution \(\nu\) on a simply connected nilpotent Lie group \(G\), the equality \(\nu*\xi = \nu*\xi'\) implies \(\xi = \xi'\). This gives the correction of a lemma in the author's paper cited above.
scientific article; zbMATH DE number 1333163
scientific article