An extension of Mercer's theory to L-p

作者：Ferreira Jose C^*; Menegatto Valdir A

来源：Positivity, 2012, 16(2): 197-212.

DOI：10.1007/s11117-011-0118-8

• Positive integral operators
• Eigenvalues
• Positive definite kernels
• Mercer's theory
• Norm estimates

Let X be a topological space, either locally compact or first countable, endowed with a strictly positive measure nu and kappa : L-2(X, nu) -> L-2(X, nu) an integral operator generated by a Mercer like kernel K. In this paper we extend Mercer's theory for K and kappa under the assumption that the function chi is an element of X -> K (chi, chi) belongs to some L-p/2(X, nu), p >= 1. In particular, we obtain series representations for K and some powers of kappa, with convergence in the p-mean, and show that the range of certain powers of kappa contains continuous functions only. These results are used to estimate the approximation numbers of a modified version of kappa acting on L-p (X, nu).

• 出版日期2012-6