We analyze some aspects of Mercer's theory when the integral operators act on L²(X,σ), where X is a first countable topological space and σ 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 σ is finite.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm126-1-9,
author = {M. H. Castro and V. A. Menegatto and A. P. Peron},
title = {Integral operators generated by Mercer-like kernels on topological spaces},
journal = {Colloquium Mathematicae},
volume = {126},
year = {2012},
pages = {125-138},
zbl = {1242.45013},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm126-1-9}
}
M. H. Castro; V. A. Menegatto; A. P. Peron. Integral operators generated by Mercer-like kernels on topological spaces. Colloquium Mathematicae, Tome 126 (2012) pp. 125-138. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm126-1-9/