单值与多值希尔伯特丛的重赋范

Measurable bundles of Banach algebras

In the present paper we investigate Banach--Kantorovich algebras over faithful solid subalgebras of algebras measurable functions. We prove that any Banach--Kantorovich algebra over faithful solid subalgebras of algebra measurable functions represented as a measurable bundle of Banach algebras with vector-valued lifting. We apply such representation to the spectrum of elements Banach--Kantorovich algebras.

向量值函数空间的单纯性

Simpliciality of vector-valued function spaces

该研究针对向量值函数空间的积分表示问题，提出向量单纯性与弱单纯性两种推广概念，分析二者的性质差异与关联，并给出更精细的表示定理及唯一性刻画，丰富了向量值函数空间理论。

关于在$L_{\infty}(Ω)$上Kaplansky-希尔伯特模块上的向量值Mercer定理

Mercer's Theorem for Vector-Valued Reproducing Kernel Hilbert Spaces in Kaplansky-Hilbert Modules over $L_{\infty}(Ω)$

本文在$L_{\infty}(Ω)$上Kaplansky-希尔伯特模块上扩展了Mercer定理，通过构造等距同构和向量值提升公理化体系，建立了偏积分算子的三个基本性质的等价性。

齐性空间上向量丛值微分形式的傅里叶变换与赫尔加松傅里叶变换

Fourier and Helgason Fourier transforms for Vector Bundle-valued Differential Forms on Homogeneous Spaces

研究齐性空间上向量丛值微分形式的傅里叶变换，利用经典傅里叶变换函数方程观点推导出赫尔加松傅里叶变换，给出了傅里叶变换的具体形式及相关公式，为调和分析在该空间的研究提供了重要方法。

Manifold-valued generalized functions in full Colombeau spaces

We introduce the notion of generalized function taking values in a smooth manifold into the setting of full Colombeau algebras. After deriving a number of characterization results we also introduce a corresponding concept of generalized vector bundle homomorphisms and, based on this, provide a definition of tangent map for such generalized functions.

关于叶状结构上顶点算子代数丛的全纯群胚

On holonomy groupoid of vertex operator algebra bundles on foliations

本文针对光滑复流形上的叶状结构，引入了截面由代数扩张$V$-模$W$-值微分提供的顶点算子代数$V$丛范畴，给出了内在坐标无关表述，并将其截面空间上同调与全纯群胚的顶点算子代数上同调等同。

Nonlinear generalized sections of vector bundles

We present an extension of J. F. Colombeau's theory of nonlinear generalized functions to spaces of generalized sections of vector bundles. Our construction builds on classical functional analytic notions, which is the key to having a canonical geometric embedding of vector bundle valued distributions into spaces of generalized sections. This permits to have tensor products, invariance under diffeomorphisms, covariant derivatives and the sheaf property. While retaining as much compatibility to L. Schwartz' theory of distributions as possible, our theory provides the basis for a rigorous and ge…

Smoothing operators for vector-valued functions and extension operators

For suitable finite-dimensional smooth manifolds M (possibly with various kinds of boundary or corners), locally convex topological vector spaces F and non-negative integers k, we construct continuous linear operators S_n from the space of F-valued k times continuously differentiable functions on M to the corresponding space of smooth functions such that S_n(f) converges to f in C^k(M,F) as n tends to infinity, uniformly for f in compact subsets of C^k(M,F). We also study the existence of continuous linear right inverses for restriction maps from C^k(M,F) to C^k(L,F) if L is a closed subset of…

The metric-valued Lebesgue differentiation theorem in measure spaces and its applications

We prove a version of the Lebesgue Differentiation Theorem for mappings that are defined on a measure space and take values into a metric space, with respect to the differentiation basis induced by a von Neumann lifting. As a consequence, we obtain a lifting theorem for the space of sections of a measurable Banach bundle and a disintegration theorem for vector measures whose target is a Banach space with the Radon-Nikodým property.

局部化算子值核在$*-$半群oid作用下的不变性

Localised Operator Valued Kernels Invariant under Actions of $*$-Semigroupoids

本文研究了在$*-$半群oid作用下保持不变的局部化算子值核，证明了其广义$*-$表示的存在性，并探讨了Hermitian核的再生核Krein空间生成条件。

Maximal accretive extensions of Schrödinger operators on vector bundles over infinite graphs

Given a Hermitian vector bundle over an infinite weighted graph, we define the Laplacian associated to a unitary connection on this bundle and study the essential self-adjointness of a perturbation of this Laplacian by an operator-valued potential. Additionally, we give a sufficient condition for the resulting Schrödinger operator to serve as the generator of a strongly continuous contraction semigroup in the corresponding l^{p}-space.

Aspects of differential calculus related to infinite-dimensional vector bundles and Poisson vector spaces

We prove various results in infinite-dimensional differential calculus which relate differentiability properties of functions and associated operator-valued functions (e.g., differentials). The results are applied in two areas: 1. in the theory of infinite-dimensional vector bundles, to construct new bundles from given ones, like dual bundles, topological tensor products, infinite direct sums, and completions (under suitable hypotheses). 2. in the theory of locally convex Poisson vector spaces, to prove continuity of the Poisson bracket and continuity of passage from a function to the associat…