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Han Ju Lee

Publications and source records attributed to Han Ju Lee.

35 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.

math.FA↗
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.

math.FA↗
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.

math.FA↗
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.

math.FA↗
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.

math.FA↗
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.

math.FA↗
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.

math.FA↗
arXiv2008

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

Using the variational method, it is shown that the set of all strong peak functions in a closed algebra $A$ of $C_b(K)$ is dense if and only if the set of all strong peak points is a norming subset of $A$. As a corollary we can induce the denseness of strong peak functions on other certain spaces. In case that a set of uniformly strongly exposed points of a Banach space $X$ is a norming subset of $\mathcal{P}({}^n X)$, then the set of all strongly norm attaining elements in $\mathcal{P}({}^n X)$ is dense. In particular, the set of all points at which the norm of $\mathcal{P}({}^n X)$ is Fréchet differentiable is a dense $G_δ$ subset. In the last part, using Reisner's graph theoretic-approach, we construct some strongly norm attaining polynomials on a CL-space with an absolute norm. Then we show that for a finite dimensional complex Banach space $X$ with an absolute norm, its polynomial numerical indices are one if and only if $X$ is isometric to $\ell_\infty^n$. Moreover, we give a characterization of the set of all complex extreme points of the unit ball of a CL-space with an absolute norm.

math.FA↗
arXiv2007

Boundaries for algebras of holomorphic functions on Banach spaces

We study the relations between boundaries for algebras of holomorphic functions on Banach spaces and complex convexity of their balls. In addition, we show that the Shilov boundary for algebras of holomorphic functions on an order continuous sequence space $X$ is the unit sphere $S_X$ if $X$ is locally c-convex. In particular, it is shown that the unit sphere of the Orlicz-Lorentz sequence space $λ_{ϕ, w}$ is the Shilov boundary for algebras of holomorphic functions on $λ_{ϕ, w}$ if $ϕ$ satisfies the $δ_2$-condition.

math.FA↗
arXiv2007

Bishop's Theorem and Differentiability of a subspace of $C_b(K)$

Let $K$ be a Hausdorff space and $C_b(K)$ be the Banach algebra of all complex bounded continuous functions on $K$. We study the Gâteaux and Fréchet differentiability of subspaces of $C_b(K)$. Using this, we show that the set of all strong peak functions in a nontrivial separating separable subspace $H$ of $C_b(K)$ is a dense $G_δ$ subset of $H$, if $K$ is compact. This gives a generalized Bishop's theorem, which says that the closure of the set of strong peak point for $H$ is the smallest closed norming subset of $H$. The classical Bishop's theorem was proved for a separating subalgebra $H$ and a metrizable compact space $K$. In the case that $X$ is a complex Banach space with the Radon-Nikodým property, we show that the set of all strong peak functions in $A_b(B_X)=\{f\in C_b(B_X) : f|_{B_X^\circ} {is holomorphic}\}$ is dense. As an application, we show that the smallest closed norming subset of $A_b(B_X)$ is the closure of the set of all strong peak points for $A_b(B_X)$. This implies that the norm of $A_b(B_X)$ is Gâteaux differentiable on a dense subset of $A_b(B_X)$, even though the norm is nowhere Fréchet differentiable when $X$ is nontrivial. We also study the denseness of norm attaining holomorphic functions and polynomials. Finally we investigate the existence of numerical Shilov boundary.

math.FA↗
arXiv2007

Notes on the geometry of space of polynomials

We show that the symmetric injective tensor product space $\hat{\otimes}_{n,s,ε}E$ is not complex strictly convex if E is a complex Banach space of $\dim E \ge 2$ and if $n\ge 2$ holds. It is also reproved that $\ell_\infty$ is finitely represented in $\hat{\otimes}_{n,s,ε}E$ if E is infinite dimensional and if $n\ge 2$ holds, which was proved in the other way by Dineen.

math.FA↗
arXiv2007

Banach spaces with polynomial numerical index 1

We characterize Banach spaces with polynomial numerical index 1 when they have the Radon-Nikodým property. The holomorphic numerical index is introduced and the characterization of the Banach space with holomorphic numerical index 1 is obtained when it has the Radon-Nikodým property.

math.FA↗
arXiv2007

Randomized series and Geometry of Banach spaces

We study some properties of the randomized series and their applications to the geometric structure of Banach spaces. For $n\ge 2$ and $1<p<\infty$, it is shown that $\ell_\infty^n$ is representable in a Banach space $X$ if and only if it is representable in the Lebesgue-Bochner $L_p(X)$. New criteria for various convexity properties in Banach spaces are also studied. It is proved that a Banach lattice $E$ is uniformly monotone if and only if its $p$-convexification $E^{(p)}$ is uniformly convex and that a Köthe function space $E$ is upper locally uniformly monotone if and only if its $p$-convexification $E^{(p)}$ is midpoint locally uniformly convex.

math.FA↗
arXiv2007

Norm and Numerical Peak Holomorphic Functions on Banach Spaces

We introduce the notion of numerical (strong) peak function and investigate the denseness of the norm and numerical peak functions on complex Banach spaces. Let $A_b(B_X:X)$ be the Banach space of all bounded continuous functions $f$ on the unit ball $B_X$ of a Banach space $X$ and their restrictions $f|_{B_X^\circ}$ to the open unit ball are holomorphic. In finite dimensional spaces, we show that the intersection of the set of all norm peak functions and the set of all numerical peak functions is a dense $G_δ$ subset of $A_b(B_X:X)$. We also prove that if $X$ is a smooth Banach space with the Radon-Nikodým property, then the set of all numerical strong peak functions is dense in $A_b(B_X:X)$. In particular, when $X=L_p(μ)$ $(1<p<\infty)$ or $X=\ell_1$, it is shown that the intersection of the set of all norm strong peak functions and the set of all numerical strong peak functions is a dense $G_δ$ subset of $A_b(B_X:X)$. In the meanwhile, we study the properties of the numerical radius of an holomorphic function and the numerical index of subspaces of $A_b(B_X:X)$. As an application, the existence and properties of numerical boundary of $A_b(B_X:X)$ are studied. Finally, the numerical peak function in $A_b(B_X:X)$ is characterized when $X=\ell_\infty^n$ and some negative results on the denseness of numerical (strong) peak holomorphic functions are given.

math.FA↗
arXiv2007

Strong peak points and denseness of strong peak functions

Let $C_b(K)$ be the set of all bounded continuous (real or complex) functions on a complete metric space $K$ and $A$ a closed subspace of $C_b(K)$. Using the variational method, it is shown that the set of all strong peak functions in $A$ is dense if and only if the set of all strong peak points is a norming subset of $A$. As a corollary we show that if $X$ is a locally uniformly convex, complex Banach space, then the set of all strong peak functions in $\mathcal{A}(B_X)$ is a dense $G_δ$ subset. Moreover if $X$ is separable, smooth and locally uniformly convex, then the set of all norm and numerical strong peak functions in $\mathcal{A}_u(B_X:X)$ is a dense $G_δ$ subset. In case that a set of uniformly strongly exposed points of a (real or complex) Banach space $X$ is a norming subset of $\mathcal{P}({}^n X)$ for some $n\ge 1$, then the set of all strongly norm attaining elements in $\mathcal{P}({}^n X)$ is dense, in particular, the set of all points at which the norm of $\mathcal{P}({}^n X)$ is Fréchet differentiable is a dense $G_δ$ subset.

math.FA↗