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We study integral operators on the space of square-integrable functions from a compact set, X, to a separableHilbert 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 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 H. Finally, when dim 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.
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The set of bounded continuous nowhere locally uniformly continuous functions is not Borel
It is known that for $$X$$ a nowhere locally compact metric space, the set of bounded continuous, nowhere locally uniformly continuous real-valued functions on $$X$$ contains a dense $$G_\delta$$ set in the space $$C_b(X)$$ of all bounded continuous real-valued functions on $$X$$ in the supremum norm. Furthermore, when $$X$$ is separable, the set of bounded continuous, nowhere locally uniformly continuous real-valued functions on $$X$$ is itself a $$G_\delta$$ set. We show that in contrast, when $$X$$ is nonseparable, this set of functions is not even a Borel set.
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- Award ID(s):
- 1856010
- PAR ID:
- 10418766
- Date Published:
- Journal Name:
- Houston journal of mathematics
- Volume:
- 48
- Issue:
- 1
- ISSN:
- 0362-1588
- Page Range / eLocation ID:
- 183 - 187
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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