arXiv subjects

Han Ju Lee

Publications and source records attributed to Han Ju Lee.

At least 19 recordsLinked to original sources

arXiv2026

The Bishop-Phelps-Bollob\'as property for the numerical radius: a Zizler-type approach

We investigate the Bishop-Phelps-Bollob\'as property for the numerical radius (BPBp-nu) through a Zizler-type perspective on the classical Bishop-Phelps-Bollob\'as property (BPBp). This approach allows us to establish two new results: the real Banach space $\ell_\infty$ satisfies the BPBp-nu, while the complex space $\ell_1 \oplus_\infty c_0$ does not. Note that the latter provides the first natural example (constructed without renorming techniques) of a Banach space where the numerical radius attaining operators are dense but the BPBp-nu fails. Along the way, we strengthen the main results of the paper [Kim et al, On the Bishop-Phelps-Bollob\'as theorem for operators and numerical radius, Studia Math., 2016] concerning the interplay between the BPBp for the pair $(X,Y)$ and the BPBp-nu for a direct sum $X\oplus Y$ of Banach spaces. We further explore the validity of the Zizler-type BPBp across different pairs of Banach spaces, and how this property relates to the classical BPBp and the BPBp-nu. Finally, we specialize our analysis to the framework of compact operators.

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arXiv2024

On various diametral notions of points in the unit ball of some vector-valued function spaces

In this article, we study the ccs-Daugavet, ccs-$Δ$, super-Daugavet, super-$Δ$, Daugavet, $Δ$, and $\nabla$ points in the unit balls of vector-valued function spaces $C_0(L, X)$, $A(K, X)$, $L_\infty(μ, X)$, and $L_1(μ, X)$. To partially or fully characterize these diametral points, we first provide improvements of several stability results under $\oplus_\infty$ and $\oplus_1$-sums shown in the literature. For complex Banach spaces, $\nabla$ points are identical to Daugavet points, and so the study of $\nabla$ points only makes sense when a Banach space is real. Consequently, we obtain that the seven notions of diametral points are equivalent for $L_\infty(μ)$ and uniform algebra when $K$ is infinite.

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arXiv2024

Remarks on the Daugavet Property for Complex Banach Spaces

In this article, we study the Daugavet property and the diametral diameter two properties in complex Banach spaces. The characterizations for both Daugavet and $Δ$-points are revisited in the context of complex Banach spaces. We also provide relationships between some variants of alternative convexity and smoothness, nonsquareness, and the Daugavet property. As a consequence, every strongly locally uniformly alternatively convex or smooth (sluacs) Banach space does not contain $Δ$-points from the fact that such spaces are locally uniformly nonsquare. We also study the convex diametral local diameter two property (convex-DLD2P) and the polynomial Daugavet property in the vector-valued function space $A(K, X)$. From an explicit computation of the polynomial Daugavetian index of $A(K, X)$, we show that the space $A(K, X)$ has the polynomial Daugavet property if and only if either the base algebra $A$ or the range space $X$ has the polynomial Daugavet property. Consequently, we obtain that the polynomial Daugavet property, the Daugavet property, the diameteral diameter two properties, and the property ($\mathcal{D}$) are equivalent for infinite-dimensional uniform algebras.

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arXiv2024

Linear structures of norm-attaining Lipschitz functions and their complements

We solve two main questions on linear structures of (non-)norm-attaining Lipschitz functions. First, we show that for every infinite metric space $M$, the set consisting of Lipschitz functions on $M$ which do not strongly attain their norm and the zero contains an isometric copy of $\ell_\infty$, and moreover, those functions can be chosen not to attain their norm as functionals on the Lipschitz-free space over $M$. Second, we prove that for every infinite metric space $M$, neither the set of strongly norm-attaining Lipschitz functions on $M$ nor the union of its complement with zero is ever a linear space. Furthermore, we observe that the set consisting of Lipschitz functions which cannot be approximated by strongly norm-attaining ones and the zero element contains $\ell_\infty$ isometrically in all the known cases. Some natural observations and spaceability results are also investigated for Lipschitz functions that attain their norm in one way but do not in another, for several norm-attainment notions considered in the literature.

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arXiv2023

Embeddings of infinite-dimensional spaces in the sets of norm-attaining Lipschitz functions

Motivated by the result of Dantas et. al. (2023) that there exist metric spaces for which the set of strongly norm-attaining Lipschitz functions does not contain an isometric copy of $c_0$, we introduce and study a weaker notion of norm-attainment for Lipschitz functions called the pointwise norm-attainment. As a main result, we show that for every infinite metric space $M$, there exists a metric space $M_0 \subseteq M$ such that the set of pointwise norm-attaining Lipschitz functions on $M_0$ contains an isometric copy of $c_0$. We also observe that there are countable metric spaces $M$ for which the set of pointwise norm-attaining Lipschitz functions contains an isometric copy of $\ell_\infty$, which is a result that does not hold for the set of strongly norm-attaining Lipschitz functions. Several new results on $c_0$-embedding and $\ell_1$-embedding into the set of strongly norm-attaining Lipschitz functions are presented as well. In particular, we show that if $M$ is a subset of an $\mathbb{R}$-tree containing all the branching points, then the set of strongly norm-attaining Lipschitz functions contains $c_0$ isometrically. As a related result, we provide an example of metric space $M$ for which the set of norm-attaining functionals on the Lipschitz-free space over $M$ cannot contain an isometric copy of $c_0$. Finally, we compare the concept of pointwise norm-attainment with the several different kinds of norm-attainment from the literature.

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arXiv2023

Daugavet and diameter two properties in Orlicz-Lorentz spaces

In this article, we study the diameter two properties (D2Ps), the diametral diameter two properties (diametral D2Ps), and the Daugavet property in Orlicz-Lorentz spaces equipped with the Luxemburg norm. First, we characterize the Radon-Nikodým property of Orlicz-Lorentz spaces in full generality by considering all finite real-valued Orlicz functions. To show this, the fundamental functions of their Köthe dual spaces defined by extended real-valued Orlicz functions are computed. We also show that if an Orlicz function does not satisfy the appropriate $Δ_2$-condition, the Orlicz-Lorentz space and its order-continuous subspace have the strong diameter two property. Consequently, given that an Orlicz function is an N-function at infinity, the same condition characterizes the diameter two properties of Orlicz-Lorentz spaces as well as the octahedralities of their Köthe dual spaces. The Orlicz-Lorentz function spaces with the Daugavet property and the diametral D2Ps are isometrically isomorphic to $L_1$ when the weight function is regular. In the process, we observe that every locally uniformly nonsquare point is not a $Δ$-point. This fact provides another class of real Banach spaces without $Δ$-points. As another application, it is shown that for Orlicz-Lorentz spaces equipped with the Luxemburg norm defined by an N-function at infinity, their Köthe dual spaces do not have the local diameter two property, and so as other (diametral) diameter two properties and the Daugavet property.

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arXiv2023

Diameter two properties and the Radon-Nikodým property in Orlicz spaces

Some necessary and sufficient conditions are found for Banach function lattices to have the Radon-Nikodým property. Consequently it is shown that an Orlicz space $L_φ$ over a non-atomic $σ$-finite measure space $(Ω, Σ,μ)$, not necessarily separable, has the Radon-Nikodým property if and only if $φ$ is an $N$-function at infinity and satisfies the appropriate $Δ_2$ condition. For an Orlicz sequence space $\ell_φ$, it has the Radon-Nikodým property if and only if $φ$ satisfies condition $Δ_2^0$. In the second part the relationships between uniformly $\ell_1^2$ points of the unit sphere of a Banach space and the diameter of the slices are studied. Using these results, a quick proof is given that an Orlicz space $L_φ$ has the Daugavet property only if $φ$ is linear, so when $L_φ$ is isometric to $L_1$. The other consequence is that the Orlicz spaces equipped with the Orlicz norm generated by $N$-functions never have local diameter two property, while it is well-known that when equipped with the Luxemburg norm, it may have that property. Finally, it is shown that the local diameter two property, the diameter two property, the strong diameter two property are equivalent in function and sequence Orlicz spaces with the Luxemburg norm under appropriate conditions on $φ$.

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arXiv2022

On the Crawford number attaining operators

We study the denseness of Crawford number attaining operators on Banach spaces. Mainly, we prove that if a Banach space has the RNP, then the set of Crawford number attaining operators is dense in the space of bounded linear operators. We also see among others that the set of Crawford number attaining operators may be dense in the space of all bounded linear operators while they do not coincide, by observing the case of compact operators when the Banach space has a 1-unconditional basis. Furthermore, we show a Bishop-Phelps-Bollobás type property for the Crawford number for certain Banach spaces, and we finally discuss some difficulties and possible problems on the topic.

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arXiv2021

Diameter two properties in some vector-valued function spaces

We introduce a vector-valued version of a uniform algebra, called the vector-valued function space over a uniform algebra. The diameter two properties of the vector-valued function space over a uniform algebra on an infinite compact Hausdorff space are investigated. Every nonempty relatively weakly open subset of the unit ball of a vector-valued function space $A(K, (X, τ))$ over an infinite dimensional uniform algebra has the diameter two, where $τ$ is a locally convex Hausdorff topology on a Banach space $X$ compatible to a dual pair. Under the assumption on $X$ being uniformly convex with norm topology $τ$ and the additional condition that $A\otimes X\subset A(K, X)$, it is shown that Daugavet points and $Δ$-points on $A(K, X)$ over a uniform algebra $A$ are the same, and they are characterized by the norm-attainment at a limit point of the Shilov boundary of $A$. In addition, a sufficient condition for the convex diametral local diameter two property of $A(K,X)$ is also provided. As a result, the similar results also hold for an infinite dimensional uniform algebra.

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arXiv2020

On various types of density of numerical radius attaining operators

In this paper, we are interested in studying two properties related to the denseness of the operators which attain their numerical radius: the Bishop-Phelps-Bollobás point and operator properties for numerical radius (BPBpp-nu and BPBop-nu, respectively). We prove that every Banach space with micro-transitive norm and second numerical index strictly positive satisfy the BPBpp-nu and that, if the numerical index of $X$ is 1, only one-dimensional spaces enjoy it. On the other hand, we show that the BPBop-nu is a very restrictive property: under some general assumptions, it holds only for one-dimensional spaces. We also consider two weaker properties, the local versions of BPBpp-nu and BPBop-nu, where the $η$ which appears in their definition does not depend just on $ε> 0$ but also on a state $(x, x^*)$ or on a numerical radius one operator $T$. We address the relation between the local BPBpp-nu and the strong subdifferentiability of the norm of the space $X$. We show that finite dimensional spaces and $c_0$ are examples of Banach spaces satisfying the local BPBpp-nu, and we exhibit an example of a Banach space with strongly subdifferentiable norm failing it. We finish the paper by showing that finite dimensional spaces satisfy the local BPBop-nu and that, if $X$ has strictly positive numerical index and has the approximation property, this property is equivalent to finite dimensionality.

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arXiv2019

On Banach spaces whose group of isometries acts micro-transitively on the unit sphere

We study Banach spaces whose group of isometries acts micro-transitively on the unit sphere. We introduce a weaker property, which one-complemented subspaces inherit, that we call uniform micro-semitransitivity. We prove a number of results about both micro-transitive and uniformly micro-semitransitive spaces, including that they are uniformly convex and uniformly smooth, and that they form a self-dual class. To this end, we relate the fact that the group of isometries acts micro-transitively with a property of operators called the pointwise Bishop-Phelps-Bollobás property and use some known results on it. Besides, we show that if there is a non-Hilbertian non-separable Banach space with uniform micro-semitransitive (or micro-transitive) norm, then there is a non-Hilbertian separable one. Finally, we show that an $L_p(μ)$ space is micro-transitive or uniformly micro-semitransitive only when $p=2$.

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arXiv2019

On some local Bishop-Phelps-Bollobás properties

We continue a line of study about some local versions of Bishop-Phelps-Bollobás type properties for bounded linear operators. We introduce and focus our attention on two of these local properties, which we call L$_{p, o}$ and L$_{o, p}$, and we explore the relation between them and some geometric properties of the underlying spaces, such as spaces having strict convexity, local uniform rotundity, and property $β$ of Lindenstrauss. At the end of the paper, we present a diagram comparing all the existing Bishop-Phelps-Bollobás type properties with each other. Some open questions are left throughout the article.

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arXiv2019

Strong subdifferentiability and local Bishop-Phelps-Bollobás properties

It has been recently presented some local versions of the Bishop-Phelps-Bollobás type property for operators. In the present article, we continue studying these properties for multilinear mappings. We show some differences between the local and uniform versions of the Bishop-Phelps-Bollobás type results for multilinear mappings, and also provide some interesting examples which shows that this study is not just a mere generalization of the linear case. We study those properties for bilinear forms on $\ell_p \times \ell_q$ using the strong subdifferentiability of the norm of the Banach space $\ell_p \hat{\otimes}_π \ell_{q}$. Moreover, we present necessary and sufficient conditions for the norm of a Banach space $Y$ to be strongly subdifferentiable through the study of these properties for bilinear mappings on $\ell_1^N \times Y$.

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arXiv2018

There is no operatorwise version of the Bishop-Phelps-Bollobás property

Given two real Banach spaces $X$ and $Y$ with dimensions greater than one, it is shown that there is a sequence $\{T_n\}_{n\in \mathbb{N}}$ of norm attaining norm-one operators from $X$ to $Y$ and a point $x_0\in X$ with $\|x_0\|=1$, such that $\|T_n(x_0)\|\longrightarrow 1$ but $\inf_{n \in \mathbb{N}} \{\mbox{dist} (x_0,\,\{x\in X: \|T_n(x)\|=\|x\|=1\})\} >0.$ This shows that a version of the Bishop-Phelps-Bollobás property in which the operator is not changed is possible only if one of the involved Banach spaces is one-dimensional.

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arXiv2018

On the pointwise Bishop--Phelps--Bollobás property for operators

We study approximation of operators between Banach spaces $X$ and $Y$ that nearly attain their norms in a given point by operators that attain their norms at the same point. When such approximations exist, we say that the pair $(X, Y)$ has the pointwise Bishop-Phelps-Bollobás property (pointwise BPB property for short). In this paper we mostly concentrate on those $X$, called universal pointwise BPB domain spaces, such that $(X, Y)$ possesses pointwise BPB property for every $Y$, and on those $Y$, called universal pointwise BPB range spaces, such that $(X, Y)$ enjoys pointwise BPB property for every uniformly smooth $X$. We show that every universal pointwise BPB domain space is uniformly convex and that $L_p(μ)$ spaces fail to have this property when $p>2$. For universal pointwise BPB range space, we show that every simultaneously uniformly convex and uniformly smooth Banach space fails it if its dimension is greater than one. We also discuss a version of the pointwise BPB property for compact operators.

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arXiv2017

A non-linear Bishop-Phelps-Bollobás type theorem

The main aim of this paper is to prove a Bishop-Phelps-Bollobás type theorem on the unital uniform algebra A_{w^*u}(B_{X^*}) consisting of all w^*-uniformly continuous functions on the closed unit ball B_{X^*} which are holomorphic on the interior of B_{X^*}. We show that this result holds for A_{w^*u}(B_{X^*}) if X^* is uniformly convex or X^* is the uniformly complex convex dual space of an order continuous absolute normed space. The vector-valued case is also studied.

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arXiv2017

$M$-ideal properties in Orlicz-Lorentz spaces

We provide explicit formulas for the norm of bounded linear functionals on Orlicz-Lorentz function spaces $Λ_{φ,w}$ equipped with two standard Luxemburg and Orlicz norms. Any bounded linear functional is a sum of regular and singular functionals, and we show that the norm of a singular functional is the same regardless of the norm in the space, while the formulas of the norm of general functionals are different for the Luxemburg and Orlicz norm. The relationship between equivalent definitions of the modular $P_{φ,w}$ generating the dual space to Orlicz-Lorentz space is discussed in order to compute the norm of a bounded linear functional on $Λ_{φ,w}$ equipped with Orlicz norm. As a consequence, we show that the order-continuous subspace of Orlicz-Lorentz space equipped with the Luxemburg norm is an $M$-ideal in $Λ_{φ,w}$, while this is not true for the space with the Orlicz norm when $φ$ is an Orlicz $N$-function not satisfying the appropriate $Δ_2$ condition. The analogous results on Orlicz-Lorentz sequence spaces are given.

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arXiv2016

The Bishop-Phelps-Bollobás point property

In this article, we study a version of the Bishop-Phelps-Bollobás property. We investigate a pair of Banach spaces $(X, Y)$ such that every operator from $X$ into $Y$ is approximated by operators which attains its norm at the same point where the original operator almost attains its norm. In this case, we say that such a pair has the Bishop-Phelps-Bollobás point property (BPBpp). We characterize uniform smoothness in terms of BPBpp and we give some examples of pairs $(X, Y)$ which have and fail this property. Some stability results are obtained about $\ell_1$ and $\ell_\infty$ sums of Banach spaces and we also study this property for bilinear mappings.

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