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Diagonals of positive semigroups
Authors:Ben de Pagter  Anton R Schep
Institution:(1) Department of Mathematics, Delft University of Technology, P.O. Box 356, 2600 AJ Delft, The Netherlands;(2) Department of Mathematics, University of South Carolina, 29208 Columbia, SC, USA
Abstract:LetE be a Dedekind complete complex Banach lattice and letD denote the diagonal projection from the spaceL r (E) onto the centerZ(E) ofE. Let {T(t)} tge0 be a positive strongly continuous semigroup of linear operators with generatorA. The first main result is that if the spectral bounds(A) equals to zero, then the functionD(T(t)) is a center valuedp-function. The second main result is that if for lambda>0 the diagonalD(R(lambda, A)) of the resolvent operatorR(lambda, A) is strictly positive, then (D(R(lambda, A))) –1 is a center valued Bernstein function. As an application of these results it follows that the order limit limgammadarr0gammaD(R(gamma,A)) exists inZ(E) and equals the order limit lim mrarrinfin D((lambdaR(lambda, A)) m ) for any lambda>0.
Keywords:47B65  47D03  47D06
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