 
Summary: Some L2
Properties of Semigroups of Measures
on Lie Groups
David Applebaum,
Probability and Statistics Department,
University of Sheffield,
Hicks Building, Hounsfield Road,
Sheffield, England, S3 7RH
email: D.Applebaum@sheffield.ac.uk
Abstract
We investigate the induced action of convolution semigroups of
probability measures on Lie groups on the L2space of Haar measure.
Necessary and sufficient conditions are given for the infinitesimal gen
erator to be selfadjoint and the associated symmetric Dirichlet form is
constructed. We show that the generated Markov semigroup is trace
class if and only if the measures have a squareintegrable density. Two
examples are studied in some depth where the spectrum can be ex
plicitly computed, these being the ntorus and Riemannian symmetric
pairs of compact type.
Key words and phrases Lie group, Lie algebra, convolution semigroup,
