Asymptotic properties of convolution operators and limits of triangular arrays on locally compact groups

Abstract

We consider a locally compact group G and a limiting measure μ of a commutative infinitesimal triangular system (c.i.t.s.) A of probability measures on G. We show, under some restrictions on G, μ or A, that μ belongs to a continuous one-parameter convolution semigroup. In particular, this result is valid for symmetric c.i.t.s. A on any locally compact group G. It is also valid for a limiting measure μ which has 'full' support on a Zariski connected F-algebraic group G, where F is a local field, and any one of the following conditions is satisfied: (1) G is a compact extension of a closed solvable normal subgroup, in particular, G is amenable, (2) μ has finite one-moment or (3) μ has density and in case the characteristic of F is positive, the radical of G is F-defined. We also discuss the spectral radius of the convolution operator of a probability measure on a locally compact group G, we show that it is always positive for any probability measure on G, and it is also multiplicative in case of symmetric commuting measures.