On the numerical index of absolute symmetric norms on the plane

On Absolutely Norm attaining Operators

We give necessary and sufficient conditions for a bounded operator defined between complex Hilbert spaces to be absolutely norm attaining. We discuss structure of such operators in the case of self-adjoint and normal operators separately. Finally, we discuss several properties of absolutely norm attaining operators.

绝对单调和多重单调函数类上的Kolmogorov问题及其与Markov矩问题的联系

Kolmogorov problem on classes of absolute monotone and multiple monotone functions and its connection with Markov moment problem

本文解决了多个单调函数和绝对单调函数类在任意范数的情况下，具有给定导数范数的函数的存在性的Kolmogorov问题。我们还展示了Kolmogorov问题与马尔可夫矩问题的联系。

S.N.上绝对赋范算子的谱特征。理想

A Spectral Characterization of Absolutely Norming Operators on S.N. Ideals

文[6]在任意维复希尔伯特空间上引入了一类绝对赋范算子，文[11]建立了这类算子的谱特征定理。本文将绝对赋范算子的概念推广到各种对称范数。我们建立了复希尔伯特空间上关于各种对称范数绝对赋范的算子的几个谱特征定理。它还表明，对于许多对称规范的绝对赋范运营商有相同的频谱表征证明较早的一类运营商是绝对赋范相对于通常的运营商规范。最后，我们证明了代数$\mathcal B (\mathcal H)$上的对称范数的存在性，即使是单位算子也不能达到它的范数。

关于规范一多项式的多项式数值指数

Polynomial numerical index with respect to a norm-one polynomial

本文提出了关于规范一多项式的多项式数值指数概念，扩展了数值指数到多项式领域，并研究了其性质和示例。

Ordered normed spaces of functions of bounded variation

In this paper, we define and study the space of all the functions of bounded variation $f:[x,y]\to \mathbb{Y}$ denoted by $\mathcal{BV}[x,y],$ where $[x,y]$ is an ordered interval and $\mathbb{Y}$ is an absolute order unit space having vector lattice structure. By default, under the order structure of $\mathbb{Y},$ the space $\mathcal{BV}[x,y]$ forms a nearer absolute order unit space structure and in some cases it turns out to be an absolute order unit space (in fact, a unital $AM$-space). By help of variation function, we also define a different kind of order structure on the space $\mathcal…

Index of symmetry and topological classification of asymmetric normed spaces

Let X, Y be asymmetric normed spaces and Lc(X, Y) the convex cone of all linear continuous operators from X to Y. It is known that in general, Lc(X, Y) is not a vector space. The aim of this note is to prove, using the Baire category theorem, that if Lc(X, Y) is a vector space for some asymmetric normed space Y , then X is isomorphic to its associated normed space (the converse is true for every asymmetric normed space Y and is easy to establish). For this, we introduce an index of symmetry of the space X denoted c(X) $\in$ [0, 1] and we give the link between the index c(X) and the fact that L…

Absolute Continuity of Function on Topological Space using Measure

The prime objective of this paper is to develop the notion of absolute continuity of functions on a more general setting outside $\R$. For this we have considered a topological space which is a measure space as well. We have built axioms for making the $σ$- algebra and measure compatible with the topology of the space. These spaces are termed as \textit{topological measure space} (in short \textit{tms}). $\R^n$ with usual topology, Lebesgue $σ$-algebra and Lebesgue measure is a relevant example of tms. Further, we have presented a new tms structure on second countable metric spaces with the de…

On the Lipschitz numerical index of Banach spaces

We provide some new consequences on the Lipschitz numerical radius and index which were introduced recently. More precisely, we give some renorming results on the Lipschitz numerical index, introduce a concept of Lipschitz numerical radius attaining functions in order to show that the denseness fails for an arbitrary Banach space, and study a Lipschitz version of Daugavet centers. Furthermore, we discuss the Lipschitz numerical index of vector-valued function spaces, absolute sums of Banach spaces, the Köthe-Bochner spaces, and Banach spaces which contain a dense union of increasing family of …

关于Lipschitz映射的绝对强暴露

On absolute strong exposure for Lipschitz maps

研究Lipschitz映射的强暴露性质，通过引入强暴露Lipschitz映射，证明其与典范线性化的绝对强暴露性等价，借此转移结果，并应用于建立相关稠密性等结论，还研究了弱序列稠密性。

Daugavet- and Delta-points in absolute sums of Banach spaces

A Daugavet-point (resp.~$Δ$-point) of a Banach space is a norm one element $x$ for which every point in the unit ball (resp.~element $x$ itself) is in the closed convex hull of unit ball elements that are almost at distance 2 from $x$. A Banach space has the well-known Daugavet property (resp.~diametral local diameter 2 property) if and only if every norm one element is a Daugavet-point (resp.~$Δ$-point). This paper complements the article "Delta- and Daugavet-points in Banach spaces" by T. A. Abrahamsen, R. Haller, V. Lima, and K. Pirk, where the study of the existence of Daugavet- and $Δ$-po…

论算子的Davis-Wielandt壳与赋范线性空间的Davis-Wielandt指数

On the Davis-Wielandt shell of an operator and the Davis-Wielandt index of a normed linear space

该研究针对赋范线性空间上算子的Davis-Wielandt壳与半径展开，证明修正后的Davis-Wielandt半径可定义与通常算子范数等价的范数，引入Davis-Wielandt指数并给出特定及一般多面体Banach空间下的计算与估计方法。

强峰值点和强范数可达点及其在稠密性和多项式数值指标中的应用

Strong peak points and strongly norm attaining points with applications to denseness and polynomial numerical indices

利用变分方法，证明了$C_b(K)$的闭代数$A$中的强峰函数集是稠密的当且仅当强峰点集是$A$的赋范子集.作为推论，我们可以导出强峰函数在其他空间上的稠密性。若巴拿赫空间$X$的一致强暴露点集是$\mathcal{P}({}^n X)$的赋范子集，则$\mathcal{P}({}^n X)$中的所有强赋范元素集是稠密的.特别地，$\mathcal{P}({}^n X)$的范数是Fréchet可微的所有点的集合是稠密的$G_δ$子集。最后，利用Reisner的图论方法，构造了CL-空间上具有绝对范数的强范数可达多项式。然后证明了对于有限维复巴拿赫空间$X$，它的多项式指数为1当且仅当$X$与$\ell_\infty^n$等距.此外，我们给出了CL-空间单位球的所有复极值点的集合的一个特征。