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Integral representations of resolvents and semigroups

Integral representations of resolvents and semigroups Abstract.The goal of this paper is to find out under which conditions the resolvent (, ) of the generator A of a positive semigroup T = (T(f))t^Q on the space LP(Q) (l < p < oo) is an integral operator. For this purpose we investigate the integral representability of some integrals of operator-valued functions (Theorems 2.1 and 2.10). 1991 Mathematics Subject Classification: 47G10; 47D03, 47B07, 47B38, 47B65, 47A10. Introduction Let c: IRN be open and l < p < oo. An operator U on LP(Q) is called an integral operator if there exists a measurable function K: x -» C such that )f(x)dx j-a.e. Using classical methods, boundary value problems are frequently solved by the use of Green's function so that the solution is obtained by an integral operator. On the other band, modern variational methods yield easily weak Solutions in much more generality. But these methods give no Information on the solution operator. The purpose of the present paper is to investigate under which conditions the resolvent R (, A) of the generator A of a positive semigroup T= (T(t))t^.0 on /() (l < p < oo) is an integral operator. The following question arises naturally. Assume http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Integral representations of resolvents and semigroups

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Publisher
de Gruyter
Copyright
Copyright © 2009 Walter de Gruyter
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.1994.6.111
Publisher site
See Article on Publisher Site

Abstract

Abstract.The goal of this paper is to find out under which conditions the resolvent (, ) of the generator A of a positive semigroup T = (T(f))t^Q on the space LP(Q) (l < p < oo) is an integral operator. For this purpose we investigate the integral representability of some integrals of operator-valued functions (Theorems 2.1 and 2.10). 1991 Mathematics Subject Classification: 47G10; 47D03, 47B07, 47B38, 47B65, 47A10. Introduction Let c: IRN be open and l < p < oo. An operator U on LP(Q) is called an integral operator if there exists a measurable function K: x -» C such that )f(x)dx j-a.e. Using classical methods, boundary value problems are frequently solved by the use of Green's function so that the solution is obtained by an integral operator. On the other band, modern variational methods yield easily weak Solutions in much more generality. But these methods give no Information on the solution operator. The purpose of the present paper is to investigate under which conditions the resolvent R (, A) of the generator A of a positive semigroup T= (T(t))t^.0 on /() (l < p < oo) is an integral operator. The following question arises naturally. Assume

Journal

Forum Mathematicumde Gruyter

Published: Jan 1, 1994

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