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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