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Perturbations in excessive structures. (English) Zbl 0534.47008
Complex analysis - Proc. 5th Rom.-Finn. Semin., Bucharest 1981, Part 2, Lect. Notes Math. 1014, 155-187 (1983).
[For the entire collection see Zbl 0516.00016.]
In the present paper the perturbation theory of operators is extended to perturbations of a resolvent \((V_{\lambda})_{\lambda>0}\) of kernels on a measurable space (E,\({\mathcal B})\) with respect to a kernel B. The B- perturbation of \((V_{\lambda})_{\lambda>0}\) is the resolvent \((W_{\lambda})_{\lambda>0}\) defined by \(W_{\lambda}=V_{\lambda}\sum^{\infty}_{n=0}(BV_{\lambda})^ n,\quad \lambda>0.\) The resolvent equation for \((W_{\lambda})\) is proved directly from the corresponding equation for \((V_{\lambda})\). The main result of the first section states that a finite non-negative measurable function s on E is \((W_{\lambda})\)-excessive if and only if it is \((V_{\lambda})\)-excessive and \(\lambda V_{\lambda}s+V_{\lambda}Bs\leq s\) for all \(\lambda>0\). The theory is also developed for semigroups of kernels. The results are applied to the case, where E is the direct sum of finitely many measurable spaces \(E_ 1,...,E_ n\), the resolvent \((V_{\lambda})\) is the direct sum of resolvents on each \(E_ i\), and the kernel B, considered as a matrix of kernels, has zeros on and above the diagonal. This case extends the triangular resolvents discussed by N. Bouleau \((n=2)\) [J. Math. Pures Appl. 59, 187-240 (1980; Zbl 0403.60068)].
Reviewer: C.Berg
MSC:
47A55 Perturbation theory of linear operators
47B38 Linear operators on function spaces (general)
60J45 Probabilistic potential theory
47A10 Spectrum, resolvent
31C99 Generalizations of potential theory