Amanollah Assadi, Hadi Haghshenas
The problems connected with equivalent norms lie at the heart of Banach space theory. This is a short survey on some recent as well as classical results and open problems in renormings of Banach spaces.
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Megrelishvili defines \emph{light groups} of isomorphisms of a Banach space as the groups on which the Weak and Strong Operator Topologies coincide, and proves that every bounded group of isomorphisms of Banach spaces with the Point of Continuity Property (PCP) is light. We investigate this concept for isomorphism groups $G$ of classical Banach spaces $X$ without the PCP, specially isometry groups, and relate it to the existence of $G$-invariant LUR or strictly convex renormings of $X$.
Remarks about symmetry-type conditions of conditional bases of Banach spaces
该研究探讨了准Banach空间中条件基的对称性和扩散性，通过重范数构造使排列和递增映射成为等距映射，扩展了经典等距重范数定理的应用范围。
We study several classical concepts in the topic of strict convexity of norms in infinite dimensional Banach spaces. Specifically, and in descending order of strength, we deal with Uniform Rotundity (UR), Weak Uniform Rotundity (WUR) and Uniform Rotundity in Every Direction (URED). Our first three results show that we may distinguish between all of these three properties in every Banach space where such renormings are possible. Specifically, we show that in every infinite dimensional Banach space which admits a WUR (resp. URED) renorming, we can find a norm with the same condition and which mo…
Quotients of continuous convex functions on nonreflexive Banach spaces
在每个非自反巴拿赫空间X上，存在正连续凸函数f使得1/f不是d.c.函数（即，两个连续凸函数的差）。这个结果与已知结果一起意味着X是自反的当且仅当两个连续凸函数的每一个处处定义的商是d.c.函数。我们的构造还给出了克利关于非自反空间和非范达到泛函的重赋结果的更强版本。
Von Neumann-Jordan Constants in Banach Tensor Products
研究Banach张量积在注入和射影范数下的von Neumann-Jordan常数，建立下界、刻画极值情形并确定重赋范包络，进而阐明张量范数、维数和重赋范对该几何常数的影响。
We strengthen the property $Δ$ of a function $f:[ω_2]^2\rightarrow [ω_2]^{\leq ω}$ considered by Baumgartner and Shelah. This allows us to consider new types of amalgamations in the forcing used by Rabus, Juhász and Soukup to construct thin-very tall compact scattered spaces. We consistently obtain spaces $K$ as above where $K^n$ is hereditarily separable for each $n\in\N$. This serves as a counterexample concerning cardinal functions on compact spaces as well as having some applications in Banach spaces: the Banach space $C(K)$ is an Asplund space of density $\aleph_2$ which has no Fréchet sm…
Variants of order semicontinuity in Banach lattices
本文研究巴拿赫格中序半连续性的变体性质，建立其关系与重赋范稳定性，证明序连续赋范格的刻画、半连续AM-空间的刻画等结论，统一简化单调完备性刻画的证明并给出定量版本。
The paper elucidates the relationship between the density of a Banach space and possible sizes of well-separated subsets of its unit sphere. For example, it is proved that for a large enough space $X$, the unit sphere $S_X$ always contains an uncountable $(1+)$-separated subset. In order to achieve this, new results concerning the existence of large Auerbach systems are established that happen to be sharp for the class of WLD spaces. In fact, we offer the first consistent example of a non-separable WLD Banach space that contains no uncountable Auerbach system, as witnessed by a renorming of $c…
The Bishop-Phelps-Bollobás property for the numerical radius: a Zizler-type approach
本文通过齐兹勒型方法研究数值半径的比什普-佩尔-博洛巴斯性质，发现实Banach空间ℓ∞满足该性质，而复空间ℓ₁⊕∞c₀不满足，并探讨了该性质与经典BPBp及数值半径达到算子的关系。
Diameter, radius and Daugavet index of thickness of slices in Banach spaces
构造具有r-BSP性质的Banach空间，其单位球切片直径下确界为1，否定Ivakhno问题；并证明Daugavet性质空间可重赋范使厚度指数大于2-ε且存在小直径切片。
Linear structures in the set of non-norm-attaining operators on Banach spaces
研究Banach空间上非范数达算子集合中的线性结构，证明非空集合在空间可性、最大空间可性及(α,β)-空间可性下的结果，扩展了已知结论。
We show that the Kottman constant $K(\cdot)$, together with its symmetric and finite variations, is continuous with respect to the Kadets metric, and they are log-convex, hence continuous, with respect to the interpolation parameter in a complex interpolation schema. Moreover, we show that $K(X)\cdot K(X^*)\geqslant 2$ for every infinite-dimensional Banach space $X$. We also consider the isomorphic Kottman constant (defined as the infimum of the Kottman constants taken over all renormings of the space) and solve the main problem left open in [CaGoPa17], namely that the isomorphic Kottman const…
First we consider the following problem which dates back to Chebyshev, Zolotarev and Achieser: among all trigonometric polynomials with given leading coefficients $a_0,...,a_l,$ $b_0,...,b_l \in \mathbb R$ find that one with least maximum norm on $[0,2 π].$ We show that the minimal polynomial is on $[0,2 π]$ asymptotically equal to a Blaschke product times a constant where the constant is the greatest singular value of the Hankel matrix associated with the $τ_j = a_j + i b_j.$ As a special case corresponding statements for algebraic polynomials follow. Finally the minimal norm of certain linea…
We establish imbedding properties between Grand Lebesgue Spaces and (generalized) Lorentz-Zygmund ones. We extend some known previous results concerning imbedding theorems between Grand Lebesgue and classical Lebesgue-Riesz spaces and we show the exactness of the obtained estimates.
The article is a survey related to a classical unsolved problem in Banach space theory, appearing in Banach's famous book in 1932, and known as the Mazur rotations problem. Although the problem seems very difficult and rather abstract, its study sheds new light on the importance of norm symmetries of a Banach space, demonstrating sometimes unexpected connections with renorming theory and differentiability in functional analysis, with topological group theory and the theory of representations, with the area of amenability, with Fraïssé theory and Ramsey theory, and led to development of concept…
In this survey article some classical results concerning real interpolation between Hardy spaces are briefly presented and then it is explained how those results can be used to establish Yano-type extrapolation theorems for Hardy spaces. Some new extensions and variants of certain classical endpoint theorems in harmonic analysis are obtained as applications of the extrapolation results presented here.
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