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Some L2 properties of semigroups of measures on Lie groups

Applebaum, D. (2009) Some L2 properties of semigroups of measures on Lie groups. Semigroup Forum, 79 (2). pp. 217-228. ISSN 0037-1912

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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.

Item Type: Article
Copyright, Publisher and Additional Information: © 2009 Springer. This is an author produced version of a paper subsequently published in Semigroup Forum. Uploaded in accordance with the publisher's self-archiving policy.
Keywords: Lie group; Lie algebra; Convolution semigroup; Hunt semigroup; Hunt generator; Dirichlet form; Beurling-Deny representation; Trace-class operator; Hilbert-Schmidt operator; Lévy-Khinchine formula; Riemannian symmetric pairs; Spherical function; Subordinator
Academic Units: The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield)
Depositing User: Miss Anthea Tucker
Date Deposited: 30 Sep 2009 11:15
Last Modified: 08 Feb 2013 16:59
Published Version: http://dx.doi.org/10.1007/s00233-008-9130-0
Status: Published
Publisher: Springer
Refereed: Yes
Identification Number: 10.1007/s00233-008-9130-0
URI: http://eprints.whiterose.ac.uk/id/eprint/9805

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