Integral operators generated by Mercer-like kernels on topological spaces

Volume 126 / 2012

M. H. Castro, V. A. Menegatto, A. P. Peron 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.

Authors

  • M. H. CastroFaculdade de Matemática
    Universidade Federal de Uberlândia
    Caixa Postal 593
    38400-902 Uberlândia MG, Brasil
    e-mail
  • V. A. MenegattoDepartamento de Matemática
    ICMC-USP - São Carlos
    Caixa Postal 668
    13560-970 São Carlos SP, Brasil
    e-mail
  • A. P. PeronDepartamento de Matemática
    ICMC-USP - São Carlos
    Caixa Postal 668
    13560-970 São Carlos SP, Brasil
    e-mail

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