Jorge Galindo

Convolution operators on locally compact groups

When G is a locally compact group and \mu\in M(G) is a bounded measure, the p-convolution operator, 1\leq p\leq \infty, defined by \mu is the bounded operator \lambda_p(\mu) \colon L^p(G)\to L^p(G) given by

    \[\lambda_p(\mu)f(x)=(\mu \ast f)(x)= \int f\left(t^{-1}x\right) \, d\mu(t)\quad x\in G.\]

The properties of these operators vary largely depending on p, the structure of G and the position of the support of \mu in G.
In this talk, we will focus on the convergence of the sequence of Cesàro means

    \[\lambda_p(\mu){[n]}= \frac{1}{n}\sum{k=1}^{n} \lambda_p(\mu)^k\]

for some of the most important operator topologies: the strong operator topology, the weak operator topology ad the operator norm topology. We will try to give a self-contained introduction to the subject that covers classical and recent results. Among the latter we will report on those obtained in joint work with Enrique Jordá (Universidad Politécnica de Valencia, Spain) and Alberto Rodíguez (currently at Universität Trier).