我们研究从紧集 X 到可分 Hilbert 空间 H 的平方可积函数空间上的积分算子。此类算子的核取值于 H 上 Hilbert–Schmidt 算子构成的理想。我们建立了核的正则性条件，使相应积分算子属于迹类。首先，我们将 Mercer 定理推广到算子值核，证明在一个附加假设下，连续、非负定且 Hermitian 对称的核在 L^2(X;H) 上定义一个迹类积分算子。其次，我们表明，对于定义在紧集上的一般算子值核，如果它作为映入 H 上迹类算子空间的映射是本质有界的，并且 Hölder 连续且 Hölder 指数大于二分之一，则该核是迹类的。最后，当 dim H < ∞ 时，我们表明在附加指数衰减假设成立的条件下，实轴上的矩阵值核也有类似结果。
英文摘要：
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 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 $\dim H < \infty$, we show that an analogous result also holds for matrix-valued kernels on the real line, provided that an additional exponential decay assumption holds.
On Matrix Valued Schrödinger Operators on the Discrete Real Line: Resolvent Boundary Values, Limiting Absorption Principle, Hölder Regularity and Dispersive Estimates
Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas (IIMAS), Universidad Nacional Autónoma de México (UNAM)（墨西哥国立自治大学应用数学与系统研究所）；Institute of Analysis and Algebra, Technische Universität Braunschweig（布伦瑞克工业大学分析与代数研究所）
Nonlinear and orbital directional regularities for set-valued mappings and applications to fixed points
Faculty of Mathematics, “Alexandru Ioan Cuza” University（亚历山德鲁·约安·库扎大学数学系）；“Octav Mayer” Institute of Mathematics, Iaşi Branch of Romanian Academy（罗马尼亚科学院雅西分院奥克塔夫·梅耶尔数学研究所）；Department of Mathematics and Statistics, Quy Nhon University（归仁大学数学与统计系）；XLIM UMR-CNRS 7252, Université de Limoges（利摩日大学XLIM UMR-CNRS 7252）
