1. Introduction. In this article, we shall present two spectral-theoretic characterizations of the property of recurrence for submarkovian semigroups. Firstly, the submarkovian and irreducible semigroup e~(-tA) is recurrent if and only if e~(-tA) is not L~1-stable, but the absorbed semigroups e~(-t(A+V)), V>0,≠0. This result is already known for the laplace operator by a result of Arendt et al. [5] and Batty [6]. Our second characterization is as follows (under additional assumptions):e~(-tA) is recurrent, if and only if w(-A+V)>0 for all V>0, V≠0, where w(-A+V) denotes the growth bound of the semigroup e~(t(-A+V)). This result is shown by different Methods for the laplacian on L~2(R~N) (N<2) by Simon [12, p. 114] and for certain functions of the Laplacian by Carmona et al. [7]. Our method gives the result in a more general setting as well as a new characterization of the property of recurrence for submarkovian semigroups.
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