Differentiable Functions and the Generators on A Hilbert-Lie Group
Authors: Erdal COŞKUN
Abstract: A convolution semigroup plays an important role İn the theory of probability measure on Lie groups. The basic problem is that one wants to express a semigroup as a Lévy-Khinckine formula. If (\mu_t)_{t\epsilonR*_+} is a continuous semigroup of probability + measures on a Hilbert-Lie group G, then we define T{\mu_t}f:=\integral f_a\mu_t(da) (f\epsilonC_u(G),t>0 It is apparent that (\mu_t)_t{t\epsilonR*_+} is a contİnuous operator semigroup on the space + C_u ( G) with the İnfinitesimal generator N. The generatİng functional A of this semigroup is defined by A := Iim_t-->0 1/t(T_{\mu_t}f(e) - f(e). We have the problem of consliuction of a subspace C{_2) ( G) of C_u ( G) such that the generatİng functional A on C{_(2)} ( G) exists. This result will be used Iater to show that the Levy-Khinchine formula holds for Hilbert-Lie groups.
Keywords: Continuous convolution semigroup, operator semigroup, Hilbert-Lie group, Lévy measure, infinitesimal generator, generating functional
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