Integral operators generated by Mercer-like kernels on topological spaces

M. H. Castro ¹   ; V. A. Menegatto ²   ; A. P. Peron ²

¹ Faculdade de Matemática Universidade Federal de Uberlândia Caixa Postal 593 38400-902 Uberlândia MG, Brasil
² Departamento de Matemática ICMC-USP - São Carlos Caixa Postal 668 13560-970 São Carlos SP, Brasil

Colloquium Mathematicum, Tome 126 (2012) no. 1, pp. 125-138

Résumé

We analyze some aspects of Mercer's theory when the integral operators act on $L^2(X,\sigma )$, where $X$ is a first countable topological space and $\sigma $ is a non-degenerate measure. We obtain results akin to the well-known Mercer's theorem and, under a positive definiteness assumption on the generating kernel of the operator, we also deduce series representations for the kernel, traceability of the operator and an integration formula to compute the trace. In this way, we upgrade considerably similar results found in the literature, in which $X$ is always metrizable and compact and the measure $\sigma $ is finite.

Détail

DOI : 10.4064/cm126-1-9

Keywords: analyze aspects mercers theory integral operators act sigma where first countable topological space sigma non degenerate measure obtain results akin well known mercers theorem under positive definiteness assumption generating kernel operator deduce series representations kernel traceability operator integration formula compute trace upgrade considerably similar results found literature which always metrizable compact measure sigma finite