arXiv subjects

Sun Kwang Kim

Publications and source records attributed to Sun Kwang Kim.

At least 19 recordsLinked to original sources

arXiv2026

Classical polynomial inequalities for quadratic forms on an octagonal sector

We establish a collection of sharp inequalities for real quadratic forms on the first-quadrant sector of a regular octagon. Starting from the complete extreme-point description of the associated polynomial unit ball, we compute the exact pointwise Bernstein function for the Euclidean gradient. One extreme curve controls the problem: its endpoint is active up to slope $1/2$, after which the maximizer follows an explicit Cardano branch. We obtain the sharp Markov constant $2\sqrt5$, the exact relative quadratic polarization constant $2$, the canonical unconditional constant $3$, and the body-relative Bohr radius $1/\sqrt3$. We also determine the optimal coefficient $\ell_q$-comparison for every $1\le q\le\infty$. The same norm-one polynomial is extremal for all these global constants. At $q=4/3$ the result is a sharp fixed-space coefficient inequality of \textit{Bohnenblust--Hille type}.

math.FA↗

arXiv2026

The unit ball of quadratic forms on an octagonal sector

Let \[\mathfrak O=\{(x,y)\in[0,1]^2:x+y\le \sqrt2\} \] be the first-quadrant sector of a regular octagon. For quadratic forms $P(x,y)=ax^2+bxy+cy^2$, we study the supremum norm over $\mathfrak O$. We obtain a complete five-region formula for the norm, according to whether the norming contact occurs at an endpoint or in the interior of one of the three radial sides. We then prove that the projection of the unit ball onto the $ac$-plane is exactly $[-1,1]^2$, compute both endpoints of every vertical section, and thereby parametrize the entire unit sphere. Finally, we characterize the extreme points of the unit ball as four explicit curves, their negatives, and four pairs of isolated points. The resulting description is fully explicit and reduces subsequent convex extremal problems on this polynomial space to four one-parameter families and finitely many isolated polynomials.

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arXiv2026

Projective norm-attainments and their implications

We show that nuclear norm-attaining operators (resp.\ polynomials) are always $w^*$-dense in the space of integral operators (resp.\ polynomials). Besides, the denseness is in norm if the predual space does not contain any isomorphic copy of $\ell_1$. We also show that there are reflexive spaces for which the set of projective norm-attaining elements does not coincide with the whole projective tensor product (which is indeed also reflexive here). Next, we show that if $Y$ is a II-polyhedral space, then every nuclear operator from an arbitrary space $X$ to $Y^*$ attains its nuclear norm. As a consequence, if $X^*$ or $Y^*$ has the approximation property, then the set of norm-attaining operators from $X^*$ to $Y^{**}$ is dense. Finally, we study proximinality results of a natural subspace of the projective tensor product and obtain an application to integral projective norm-attaining tensors which solves a proposed open question.

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arXiv2026

The Bishop-Phelps-Bollobás property for the numerical radius: a Zizler-type approach

We investigate the Bishop-Phelps-Bollobás property for the numerical radius (BPBp-nu) through a Zizler-type perspective on the classical Bishop-Phelps-Bollobás 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-Bollobás 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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arXiv2025

Geometry of the space of compact operators endowed with the numerical radius norm

We investigate the space of bounded linear operators on a Banach space equipped with a norm which is equivalent to the operator norm such that the subspace of compact operators is an M-ideal. In particular, we observe that the space of compact operators on $\ell_p$ $(1<p<\infty)$ equipped with the numerical radius norm is an M-ideal whenever the numerical index of $\ell_p$ is not $0$. On the other hand, we show that the space of compact operators on a Banach space containing an isomorphic copy of $\ell_1$ whose numerical index is greater than $1/2$ is not M-ideals. We also study the proximinality, the existence of farthest points and the compact perturbation property for the numerical radius.

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arXiv2024

M-ideals of compact operators and Norm attaining operators

We investigate M-ideals of compact operators and two distinct properties in norm-attaining operator theory related with M-ideals of compact operators called the weak maximizing property and the compact perturbation property. For Banach spaces $X$ and $Y$, it is previously known that if $\mathcal{K}(X,Y)$ is an M-ideal or $(X,Y)$ has the weak maximizing property, then $(X,Y)$ has the adjoint compact perturbation property. We see that their converses are not true, and the condition that $\mathcal{K}(X,Y)$ is an M-ideal does not imply the weak maximizing property, nor vice versa. Nevertheless, we see that all of these are closely related to property $(M)$, and as a consequence, we show that if $\mathcal{K}(\ell_p,Y)$ $(1<p<\infty)$ is an M-ideal, then $(\ell_p,Y)$ has the weak maximizing property. We also prove that $(\ell_1,\ell_1)$ does not have the adjoint compact perturbation property, and neither does $(\ell_1,Y)$ for an infinite dimensional Banach space $Y$ without an isomorphic copy of $\ell_1$ if $Y$ does not have the local diameter 2 property. As a consequence, we show that if $Y$ is an infinite dimensional Banach space such that $\mathcal{L}(\ell_1,Y)$ is an M-ideal, then it has the local diameter 2 property. Furthermore, we also studied various geometric properties of Banach spaces such as the Opial property with moduli of asymptotic uniform smoothness and uniform convexity.

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arXiv2022

Weak-star quasi norm attaining operators

For Banach spaces $X$ and $Y$, a bounded linear operator $T\colon X \longrightarrow Y^*$ is said to weak-star quasi attain its norm if the $σ(Y^*,Y)$-closure of the image by $T$ of the unit ball of $X$ intersects the sphere of radius $\|T\|$ centred at the origin in $Y^*$. This notion is inspired by the quasi-norm attainment of operators introduced and studied in \cite{CCJM}. As a main result, we prove that the set of weak-star quasi norm attaining operators is dense in the space of bounded linear operators regardless of the choice of the Banach spaces, furthermore, that the approximating operator can be chosen with additional properties. This allows us to distinguish the properties of weak-star quasi norm attaining operators from those of quasi norm attaining operators. It is also shown that, under certain conditions, weak-star quasi norm attaining operators share numbers of equivalent properties with other types of norm attaining operators, but that there are also a number of situations in which they behave differently from the others.

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arXiv2022

On a set of norm attaining operators and the strong Birkhoff-James orthogonality

Continuing the study of recent results on the Birkhoff-James orthogonality and the norm attainment of operators, we introduce a property namely the adjusted Bhatia-Šemrl property for operators which is weaker than the Bhatia-Šemrl property. The set of operators with the adjusted Bhatia-Šemrl property is contained in the set of norm attaining ones as it was in the case of the Bhatia-Šemrl property. It is known that the set of operators with the Bhatia-Šemrl property is norm-dense if the domain space $X$ of the operators has the Radon-Nikodým property like finite dimensional spaces, but it is not norm-dense for some classical spaces such as $c_0$, $L_1[0,1]$ and $C[0,1]$. In contrast with the Bhatia-Šemrl property, we show that the set of operators with the adjusted Bhatia-Šemrl property is norm-dense when the domain space is $c_0$ or $L_1[0,1]$. Moreover, we show that the set of functionals having the adjusted Bhatia-Šemrl property on $C[0,1]$ is not norm-dense but such a set is weak-$*$-dense in $C(K)^*$ for any compact Hausdorff $K$.

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arXiv2021

The Birkhoff-James orthogonality and norm attainment for multilinear maps

Very recently, motivated by the result of Bhatia and Šemrl which characterizes the Birkhoff-James orthogonality of operators on a finite dimensional Hilbert space in terms of norm attaining points, the Bhatia-Šemrl property was introduced. The main purpose of this article is to study the denseness of the set of multilinear maps with the Bhatia-Šemrl property which is contained in the set of norm attaining ones. Contrary to the most of previous results which were shown for operators on real Banach spaces, we prove the denseness for multilinear maps on some complex Banach spaces. We also show that the denseness of operators does not hold when the domain space is $c_0$ for arbitrary range. Moreover, we find plenty of Banach spaces $Y$ such that only the zero operator has the Bhatia-Šemrl property in the space of operators from $c_0$ to $Y$.

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arXiv2020

The Bishop-Phelps-Bollobás property on the space of $c_0$-sum

The main purpose of this paper is to study Bishop-Phelps-Bollobás type properties on $c_0$ sum of Banach spaces. Among other results, we show that the pair $(c_0(X),Y)$ has the Bishop-Phelps-Bollobás property (in short, BPBp) for operators whenever $X$ is uniformly convex and $Y$ is (complex) uniformly convex. We also prove that the pair $(c_0(X),c_0(X))$ has the BPBp for bilinear forms whenever $X$ is both uniformly convex and uniformly smooth. These extend the previously known results that $(c_0,Y)$ has the BPBp for operators whenever $Y$ is uniformly convex and $(c_0,c_0)$ has the BPBp for bilinear forms. We also obtain some results on a local BPBp which is called $\mathbf{L}_{p,p}$ for both operators and bilinear forms.

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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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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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