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Sun Kwang Kim

Publications and source records attributed to Sun Kwang Kim.

27 records · Page 2Linked to original sources
arXiv2016

On a second numerical index for Banach spaces

We introduce a second numerical index for real Banach spaces with non-trivial Lie algebra, as the best constant of equivalence between the numerical radius and the quotient of the operator norm modulo the Lie algebra. We present a number of examples and results concerning absolute sums, duality, vector-valued function spaces\ldots which show that, in many cases, the behaviour of this second numerical index differs from the one of the classical numerical index. As main results, we prove that Hilbert spaces have second numerical index one and that they are the only spaces with this property among the class of Banach spaces with one-unconditional basis and non-trivial Lie algebra. Besides, an application to the Bishop-Phelps-Bollobás property for numerical radius is given.

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arXiv2014

The Bishop-Phelps-Bollobás version of Lindenstrauss properties A and B

We study a Bishop-Phelps-Bollobás version of Lindenstrauss properties A and B. For domain spaces, we study Banach spaces $X$ such that $(X,Y)$ has the Bishop-Phelps-Bollobás property (BPBp) for every Banach space $Y$. We show that in this case, there exists a universal function $η_X(\varepsilon)$ such that for every $Y$, the pair $(X,Y)$ has the BPBp with this function. This allows us to prove some necessary isometric conditions for $X$ to have the property. We also prove that if $X$ has this property in every equivalent norm, then $X$ is one-dimensional. For range spaces, we study Banach spaces $Y$ such that $(X,Y)$ has the Bishop-Phelps-Bollobás property for every Banach space $X$. In this case, we show that there is a universal function $η_Y(\varepsilon)$ such that for every $X$, the pair $(X,Y)$ has the BPBp with this function. This implies that this property of $Y$ is strictly stronger than Lindenstrauss property B. The main tool to get these results is the study of the Bishop-Phelps-Bollobás property for $c_0$-, $\ell_1$- and $\ell_\infty$-sums of Banach spaces.

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arXiv2014

On Banach spaces with the approximate hyperplane series property

We present a sufficient condition for a Banach space to have the approximate hyperplane series property (AHSP) which actually covers all known examples. We use this property to get a stability result to vector-valued spaces of integrable functions. On the other hand, the study of a possible Bishop-Phelps-Bollobás version of a classical result of V. Zizler leads to a new characterization of the AHSP for dual spaces in terms of $w^*$-continuous operators and other related results.

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arXiv2014

Simultaneously continuous retraction and Bishop-Phelps-Bollobás type theorem

We study the existence of a retraction from the dual space $X^*$ of a (real or complex) Banach space $X$ onto its unit ball $B_{X^*}$ which is uniformly continuous in norm topology and continuous in weak-$*$ topology. Such a retraction is called a uniformly simultaneously continuous retraction. It is shown that if $X$ has a normalized unconditional Schauder basis with unconditional basis constant 1 and $X^*$ is uniformly monotone, then a uniformly simultaneously continuous retraction from $X^*$ onto $B_{X^*}$ exists. It is also shown that if $\{X_i\}$ is a family of separable Banach spaces whose duals are uniformly convex with moduli of convexity $δ_i(\varepsilon)$ such that $\inf_i δ_i(\varepsilon)>0$ and $X= \left[\bigoplus X_i\right]_{c_0}$ or $X=\left[\bigoplus X_i\right]_{\ell_p}$ for $1\le p<\infty$, then a uniformly simultaneously continuous retraction exists from $X^*$ onto $B_{X^*}$. The relation between the existence of a uniformly simultaneously continuous retraction and the Bishsop-Phelps-Bollobás property for operators is investigated and it is proved that the existence of a uniformly simultaneously continuous retraction from $X^*$ onto its unit ball implies that a pair $(X, C_0(K))$ has the Bishop-Phelps-Bollobás property for every locally compact Hausdorff spaces $K$. As a corollary, we prove that $(C_0(S), C_0(K))$ has the Bishop-Phelps-Bollobás property if $C_0(S)$ and $C_0(K)$ are the spaces of all real-valued continuous functions vanishing at infinity on locally compact metric space $S$ and locally compact Hausdorff space $K$ respectively.

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arXiv2014

On the Bishop-Phelps-Bollobás property for numerical radius

We study the Bishop-Phelps-Bollobás property for numerical radius (in short, BPBp-nu) and find sufficient conditions for Banach spaces ensuring the BPBp-nu. Among other results, we show that $L_1(μ)$-spaces have this property for every measure $μ$. On the other hand, we show that every infinite-dimensional separable Banach space can be renormed to fail the BPBp-nu. In particular, this shows that the Radon-Nikodým property (even reflexivity) is not enough to get BPBp-nu.

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arXiv2013

The Bishop-Phelps-Bollobás property for operators between spaces of continuous functions

We show that the space of bounded and linear operators between spaces of continuous functions on compact Hausdorff topological spaces has the Bishop-Phelps-Bollobás property. A similar result is also proved for the class of compact operators from the space of continuous functions vanishing at infinity on a locally compact and Hausdorff topological space into a uniformly convex space, and for the class of compact operators from a Banach space into a predual of an $L_1$-space.

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arXiv2013

The Bishop-Phelps-Bollobás theorem for operators on $L_1(μ)$

In this paper we show that the Bishop-Phelps-Bollobás theorem holds for $\mathcal{L}(L_1(μ), L_1(ν))$ for all measures $μ$ and $ν$ and also holds for $\mathcal{L}(L_1(μ),L_\infty(ν))$ for every arbitrary measure $μ$ and every localizable measure $ν$. Finally, we show that the Bishop-Phelps-Bollobás theorem holds for two classes of bounded linear operators from a real $L_1(μ)$ into a real $C(K)$ if $μ$ is a finite measure and $K$ is a compact Hausdorff space. In particular, one of the classes includes all Bochner representable operators and all weakly compact operators.

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