Integral operators with infinitely smooth bi-Carleman kernels of Mercer type

TL;DR

This paper introduces a new class of infinitely smooth bi-Carleman kernels of Mercer type, extending classical expansion theorems and characterizing operators unitarily equivalent to integral operators with these kernels.

Contribution

It defines and studies $K^ abla$ kernels of Mercer type, extending expansion theorems to non-Hermitian cases and characterizing operators unitarily equivalent to such integral operators.

Findings

Established an expansion theorem for $K^ abla$ kernels of Mercer type.

Proved that any bi-integral operator is unitarily equivalent to an integral operator with a $K^ abla$ kernel.

Extended Mercer's and Kadota's theorems to a non-Hermitian setting.

Abstract

With the aim of applications to solving general integral equations, we introduce and study in this paper a special class of bi-Carleman kernels on $\mathbb{R}\times\mathbb{R}$, called $K^\infty$ kernels of Mercer type, whose property of being infinitely smooth is stable under passage to certain left and right multiples of their associated integral operators. An expansion theorem in absolutely and uniformly convergent bilinear series concerning kernels of this class is proved extending to a general non-Hermitian setting both Mercer's and Kadota's Expansion Theorems for positive definite kernels. Another theorem proved in this paper identifies families of those bounded operators on a separable Hilbert space $\mathcal{H}$ that can be simultaneously transformed by the same unitary equivalence transformation into bi-Carleman integral operators on $L^2(\mathbb{R})$, whose kernels are $K^\infty$ kernels of Mercer type; its singleton version implies in particular that any bi-integral operator is unitarily equivalent to an integral operator with such a kernel.