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

We study integral operators on the space of square-integrable functions from a compact set, X,  to a separable Hilbert space, $$\textsf{H}.$$ The kernel of such an operator takes values in the ideal of Hilbert–Schmidt operators on $$\textsf{H}.$$ 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 $$L^2(X;\textsf{H})$$ under an additional assumption. Second, we show that a general operator-valued kernel that is defined on a compact set and that is Hölder continuous with Hölder 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 $$\textsf{H}.$$ Finally, when $$\dim \textsf{H}< \infty ,$$ we show that an analogous result also holds for matrix-valued kernels on the real line, provided that an additional exponential decay assumption holds.