Convexity of Chebyshev Sets in Hilbert Spaces

Truncated Hilbert Transform: Uniqueness and a Chebyshev series Expansion Approach

We derive a stronger uniqueness result if a function with compact support and its truncated Hilbert transform are known on the same interval by using the Sokhotski-Plemelj formulas. To find a function from its truncated Hilbert transform, we express them in the Chebyshev polynomial series and then suggest two methods to numerically estimate the coefficients. We present computer simulation results to show that the extrapolative procedure numerically works well.

Noncommutative Chebyshev inequality involving the Hadamard product

We present several operator extensions of the Chebyshev inequality for Hilbert space operators. The main version deals with the synchronous Hadamard property for Hilbert space operators. Among other inequalities, it is shown that if ${\mathfrak A}$ is a $C^*$-algebra, $T$ is a compact Hausdorff space equipped with a Radon measure $μ$ as a totaly order set, then \begin{align*} \int_{T} α(s) dμ(s)\int_{T}α(t)(A_t\circ B_t) dμ(t)\geq\Big{(}\int_{T}α(t) (A_tm_{r,α} B_t) dμ(t)\Big{)}\circ\Big{(}\int_{T}α(s) (A_sm_{r,1-α} B_s) dμ(s)\Big{)}, \end{align*} where $α\in[0,1]$, $r\in[-1,1]$ and $(A_t)_{t\…

Differentiability of Distance Function and The Proximinal Condition implying Convexity

We establish a necessary and sufficient condition for the differentiability of the distance function generated by a nonempty closed set K in a real normed linear space X under a proximinality condition on K. We do not assume the uniform differentiability constraints on the norm of the space as in Giles [16]. Hence, our result advances that of Giles [16]. We prove that the proximinal condition of Giles [16] is true for almost suns. The proximinal condition ensures convexity of an almost sun in some class of strongly smooth spaces under a differentiability condition of the distance function. A n…

Chebyshev type inequalities for Hilbert space operators

We establish several operator extensions of the Chebyshev inequality. The main version deals with the Hadamard product of Hilbert space operators. More precisely, we prove that if $\mathscr{A}$ is a $C^*$-algebra, $T$ is a compact Hausdorff space equipped with a Radon measure $μ$, $α: T\rightarrow [0, +\infty)$ is a measurable function and $(A_t)_{t\in T}, (B_t)_{t\in T}$ are suitable continuous fields of operators in ${\mathscr A}$ having the synchronous Hadamard property, then \begin{align*} \int_{T} α(s) dμ(s)\int_{T}α(t)(A_t\circ B_t) dμ(t)\geq\left(\int_{T}α(t) A_t dμ(t)\right)\circ\left(…

Uniformly convex subsets of the Hilbert space with modulus of convexity of the second order

We prove that in the Hilbert space every uniformly convex set with modulus of convexity of the second order at zero is an intersection of closed balls of fixed radius. We also obtain an estimate of this radius.

Bregman distances and Chebyshev sets

A closed set of a Euclidean space is said to be Chebyshev if every point in the space has one and only one closest point in the set. Although the situation is not settled in infinite-dimensional Hilbert spaces, in 1932 Bunt showed that in Euclidean spaces a closed set is Chebyshev if and only if the set is convex. In this paper, from the more general perspective of Bregman distances, we show that if every point in the space has a unique nearest point in a closed set, then the set is convex. We provide two approaches: one is by nonsmooth analysis; the other by maximal monotone operator theory. …

Wasserstein重心凸性检测Hilbert几何

Wasserstein Barycenter Convexity Detects Hilbertian Geometry

证明Boltzmann熵在Wasserstein重心处的凸性足以区分Hilbert空间与一般Banach空间，从而提供内蕴的最优传输测试。

On Local Convexity Of Nonlinear Mappings Between Banach Spaces

We find conditions for a smooth nonlinear map $f:U\rightarrow V$ between open subsets of Hilbert or Banach spaces to be locally convex in the sense that for some $c$ and each positive $\varepsilon<c$ the image $% f(B_\varepsilon(x))$ of each $\varepsilon$-ball $B_\varepsilon(x)\subset U$ is convex. We give a lower bound on $c$ via the second order Lipschitz constant $\mathrm{Lip}_2(f)$, the Lipschitz-open constant $\mathrm{Lip}_o(f)$ of $f$, and the 2-convexity number $\mathrm{conv}_2(X)$ of the Banach space $% X$.

Polyak's theorem on Hilbert spaces

We extend to infinite dimensional Hilbert spaces a celebrated result, due to B. Polyak, about the convexity of the joint image of quadratic functions. We give sufficient conditions which assure that the joint image is also closed. However, we show that, in general, the closedness part of Polyak's theorem does not hold in the infinite dimensional setting, even for quadratic functions generated by compact operators. We give some applications to S-lemma type results.

希尔伯特空间与半希尔伯特空间中算子的联合$q$-数值域

Joint $q$-Numerical Ranges of Operators in Hilbert and Semi-Hilbert Spaces

该研究提出并探究希尔伯特空间中有界线性算子组的$q$-数值域，建立相关$q$-数值半径不等式，还将其拓展至半希尔伯特空间并证明凸性，给出相关结果。

矩阵凸性与Toeplitz压缩算子元组的酉膨胀

Matrix convexity and unitary power dilations of Toeplitz-contractive operator tuples

本文基于Ando和Gurvits的工作，将Halmos定理推广到满足矩阵正性条件的Toeplitz压缩算子d元组，刻画了该条件，定义了新的距离度量，并证明了矩阵凸集包含的缩放常数存在。

On the ${\Ext}^2$-problem for Hilbert spaces

We show that $\Ext^2(\ell_2, \ell_2)\neq 0$ in the category of Banach spaces. This solves a sharpened version of Palamodov's problem and provides a solution to the second order version of Palais problem. We also show that $\Ext^2(\ell_1, \K)\neq 0$ in the category of quasi Banach spaces which solves the four-space problem for local convexity.

关于复合算子Berezin范围的凸性及相关问题

On the convexity of the Berezin range of composition operators and related questions

本文研究了再生核Hilbert空间上算子Berezin范围的凸性，刻画了Fock空间和Dirichlet空间上复合算子Berezin范围的凸性，并证明了二维空间上酉等价Berezin范围的椭圆范围定理类似结论。

Hilbert空间上的无界反线性算子

Unbounded Antilinear Operators on Hilbert Spaces

本文引入Hilbert空间上的无界反线性算子，建立其基本理论，包括闭值域定理、极分解定理和数值范围的凸性，并给出反线性正规算子的新结果及反线性次正规算子存在最小反线性正规扩张的充要条件，最后通过Schur和二次补框架刻画了纯反线性分块算子矩阵的可闭性。

Convexity of the Berezin Range

This paper discusses the convexity of the range of the Berezin transform. For a bounded operator $T$ acting on a reproducing kernel Hilbert space $H$ (on a set $X$), this is the set $B(T) : = \{ < Tk_x, k_x >_H : x \in X \}$, where $k_x$ is the normalized reproducing kernel for $H$ at $x \in X$. Primarily, we focus on characterizing convexity of this range for a class of composition operators acting on the Hardy space of the unit disk.

1-余弦的亚线性增长与均匀凸性

Sublinear growth of 1-cocycles and uniform convexity

研究1-余弦在有限生成群上的亚线性增长特性，证明其与Banach空间的均匀凸性相关联。

Operator extensions of Hua's inequality

We give an extension of Hua's inequality in pre-Hilbert $C^*$-modules without using convexity or the classical Hua's inequality. As a consequence, some known and new generalizations of this inequality are deduced. Providing a Jensen inequality in the content of Hilbert $C^*$-modules, another extension of Hua's inequality is obtained. We also present an operator Hua's inequality, which is equivalent to operator convexity of given continuous real function.

有限秩算子的Berezin范围的凸性

Convexity of the Berezin range of finite rank operators

本文研究Hardy空间和Bergman空间上有限秩算子的Berezin范围的凸性，利用标量不等式推导算子不等式，并讨论再生核Hilbert空间算子数值范围闭包与其Berezin范围凸包的关系。

通过一类半范数研究Berezin值域的凸性与Berezin半径不等式

On the convexity of Berezin range and Berezin radius inequalities via a class of semi-norms

该研究引入新的$\u03c3_\u03bc$-Berezin半范数并推导相关不等式，以此改进有界线性算子的Berezin半径估计，同时研究两类空间上特定算子Berezin值域的凸性。

复合算子、其 Berezin 值域的凸性及相关问题

Composition operators, convexity of their Berezin range and related questions

本文研究再生核 Hilbert 空间中复合算子 Berezin 值域的凸性，刻画 Hardy 与 Bergman 空间情形，并证明正自伴算子在超二次函数下的 Berezin 集映射定理。

关于瓦瑟斯坦空间中不同凸性概念下近端算子的稳定性

On the stability of proximal operators in Wasserstein spaces under different notions of convexity

研究瓦瑟斯坦空间中不同凸性概念下近端算子的稳定性，通过对各种泛函凸性进行探讨，特别关注其非扩张性，以深入理解该算子在瓦瑟斯坦空间中的收缩性质。

Berezin范围的凸性及Berezin半径不等式通过一类半范数

Convexity of Berezin Range and Berezin Radius Inequalities via a class of Seminorm

本文通过一类半范数研究了Berezin范围的凸性及Berezin半径不等式，改进了现有结果。

凸性、多重次调和性以及 Banach 空间中的强最大模原理

Convexity, plurisubharmonicity and the strong maximum modulus principle in Banach spaces

本文明确了凸性与多重次调和性的类比，引入严格多重次调和性概念以研究 Banach 空间中的强最大模原理，并应用于 $L^p$ 直积分给出了统一的新证明。

B-convex operator spaces

The notion of B-convexity for operator spaces, which a priori depends on a set of parameters indexed by $Σ$, is defined. Some of the classical characterizations of this geometric notion for Banach spaces are studied in this new context. For instance, an operator space is $B_Σ$-convex if and only if it has $Σ$-subtype. The class of uniformly non-$L^1(Σ)$ operator spaces, which is also the class of $B_Σ$-convex operator spaces, is introduced. Moreover, an operator space having non-trivial $Σ$-type is $B_Σ$-convex. However, the converse is false. The row and column operator spaces are nice counte…

New Geometric Constant Related to the P-angle Function in Banach Spaces

In this paper, combined with the P-angle function of Banach spaces and the geometric constants that can characterize Hilbert spaces, the new angular geometric constant is defined. Firstly, this paper explores the basic properties of the new constant and obtains some inequalities with significant geometric constants. Then according to the derived inequalities, this paper studies the relationship between the new constant and the geometric properties of Banach spaces. Furthermore, the necessary and sufficient condition for uniform non-squareness, and the sufficient conditions for uniform convexit…

谱包含数值范围边界的算子的不变子空间

Invariant subspaces for operators with spectrum containing the boundary of the numerical range

本文证明当有界希尔伯特空间算子的谱包含其数值范围的拓扑边界时，该算子存在非平凡不变子空间，并给出变体及在次正规、余次正规和Toeplitz算子中的应用。

Local characterization of strongly convex sets

Strongly convex sets in Hilbert spaces are characterized by local properties. One quantity which is used for this purpose is a generalization of the modulus of convexity δ_Ωof a set Ω. We also show that \lim_{ε\to 0} δ_Ω(ε)/ε^2 exists whenever Ωis closed and convex.

On C*-Extreme Maps and *-Homomorphisms of a Commutative C*-Algebra

The generalized state space of a commutative C*-algebra, denoted S_H(C(X)), is the set of positive unital maps from C(X) to the algebra B(H) of bounded linear operators on a Hilbert space H. C*-convexity is one of several non-commutative analogs of convexity which have been discussed in this context. In this paper we show that a C*-extreme point of S_H(C(X)) satisfies a certain spectral condition on the operators in the range of the associated positive operator-valued measure. This result enables us to show that C*-extreme maps from C(X) into K^+, the algebra generated by the compact and scala…

Locally nonconical convexity

There is a hierarchy of structure conditions for convex sets. In this paper we study a recently defined [3, 8, 9] condition called locally nonconical convexity (abbreviated LNC). Is is easy to show that every strictly convex set is LNC, as are half-spaces and finite intersections of sets of either of these types, but many more sets are LNC. For instance, every zonoid (the range of a nonatomic vector-valued measure) is LNC (Corollary 34). However, there are no infinite-dimensional compact LNC sets (Theorem 23). The LNC concept originated in a search for continuous sections, and the present pape…

保持结构的凸Lipschitz函数在Hilbert空间上从有限样本的重建

Structure-Preserving Reconstruction of Convex Lipschitz Functionals on Hilbert Spaces from Finite Samples

研究在Hilbert空间上如何通过有限样本重建凸Lipschitz函数，提出了一种保持凸性和Lipschitz正则性的显式公式，并引入了凸神经函数（CNFs）作为可训练架构。

希尔伯特空间中保持一致距离的基尔施布劳恩扩张

Kirszbraun extensions preserving uniform distance in Hilbert spaces

研究希尔伯特空间中基尔施布劳恩扩张保持一致距离的条件，证明相关等价条件，给出有限维目标应用及无限维或特定维度关系下的一些结论，如数据处理特征、提升定理和不等式等。