Integral operators generated by Mercer-like kernels on topological spaces
Volume 126 / 2012
Colloquium Mathematicum 126 (2012), 125-138
MSC: 45P05, 45H05, 47B34, 47G10, 42A82.
DOI: 10.4064/cm126-1-9
Abstract
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.
