Abstract
We investigate the induced action of convolution semigroups of probability measures on Lie groups on the L 2-space of Haar measure. Necessary and sufficient conditions are given for the infinitesimal generator to be self-adjoint 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 square-integrable density. Two examples are studied in some depth where the spectrum can be explicitly computed, these being the n-torus and Riemannian symmetric pairs of compact type.
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Communicated by Joachim Hilgert.
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Applebaum, D. Some L 2 properties of semigroups of measures on Lie groups. Semigroup Forum 79, 217–228 (2009). https://doi.org/10.1007/s00233-008-9130-0
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DOI: https://doi.org/10.1007/s00233-008-9130-0