Abstract
We consider Markov semigroups on the cone of positive finite measures on a complete separable metric space. Such a semigroup extends to a semigroup of linear operators on the vector space of measures that typically fails to be strongly continuous for the total variation norm. First we characterise when the restriction of a Markov semigroup to an invariant L 1-space is strongly continuous. Aided by this result we provide several characterisations of the subspace of strong continuity for the total variation norm. We prove that this subspace is a projection band in the Banach lattice of finite measures, and consequently obtain a direct sum decomposition.
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Communicated by Rainer Nagel.
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Hille, S.C., Worm, D.T.H. Continuity properties of Markov semigroups and their restrictions to invariant L 1-spaces. Semigroup Forum 79, 575–600 (2009). https://doi.org/10.1007/s00233-009-9176-7
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DOI: https://doi.org/10.1007/s00233-009-9176-7