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, 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 L²(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 H < ∞, we show that an analogous result also holds for matrix-valued kernels on the real line, provided that an additional exponential decay assumption holds.