A regularity condition under which integral operators with operator-valued kernels are trace class

TL;DR

This paper establishes conditions under which integral operators with operator-valued kernels are trace class, extending Mercer's theorem and providing regularity criteria based on continuity, boundedness, and decay properties.

Contribution

It extends Mercer's theorem to operator-valued kernels and introduces new regularity conditions ensuring trace class properties of such integral operators.

Findings

Continuous, nonnegative-definite kernels define trace class operators under additional assumptions.

Hölder continuity with exponent > 1/2 and boundedness implies trace class property.

Finite-dimensional case with decay conditions also yields trace class operators.

Abstract

We study integral operators on the space of square-integrable functions from a compact set, $X$, to a separable Hilbert space, $H$. The kernel of such an operator takes values in the ideal of Hilbert-Schmidt operators on $H$. We establish regularity conditions on the kernel under which the associated integral operator is trace class. First, we extend Mercer's theorem to operator-valued kernels by proving that a continuous, nonnegative-definite, Hermitian symmetric kernel defines a trace class integral operator on $L^2(X;H)$ under an additional assumption. Second, we show that a general operator-valued kernel that is defined on a compact set and that is H\"older continuous with H\"older exponent greater than a half is trace class provided that the operator-valued kernel is essentially bounded as a mapping into the space of trace class operators on $H$. Finally, when $\dim H < \infty$, we show that an analogous result also holds for matrix-valued kernels on the real line, provided that an additional exponential decay assumption holds.