arXiv subjects

Yun Sung Choi

Publications and source records attributed to Yun Sung Choi.

13 recordsLinked to original sources

arXiv2021

The spectra of Banach algebras of holomorphic functions on polydisk type domains

R.M. Aron et al. proved that the Cluster Value Theorem in the infinite dimensional Banach space setting holds for the Banach algebra $\mathcal{H}^\infty (B_{c_0})$. On the other hand, B.J. Cole and T.W. Gamelin showed that $\mathcal{H}^\infty (\ell_2 \cap B_{c_0})$ is isometrically isomorphic to $\mathcal{H}^\infty (B_{c_0})$ in the sense of an algebra. Motivated by this work, we are interested in a class of open subsets $U$ of a Banach space $X$ for which $\mathcal{H}^\infty (U)$ is isometrically isomorphic to $\mathcal{H}^\infty (B_{c_0})$. We prove that there exist polydisk type domains $U$ of any infinite dimensional Banach space $X$ with a Schauder basis such that $\mathcal{H}^\infty (U)$ is isometrically isomorphic to $\mathcal{H}^\infty (B_{c_0})$, which generalizes the result by Cole and Gamelin. Furthermore, we study the analytic and algebraic structure of the spectrum of $\mathcal{H}^\infty (U)$ and show that the Cluster Value Theorem is true for $\mathcal{H}^\infty (U)$.

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arXiv2021

Boundaries for Gelfand transform images of Banach algebras of holomorphic functions

Let $\mathcal{A}$ be a Banach algebra of bounded holomorphic functions on the open unit ball $B_X$ of a complex Banach space $X$. Considering the Gelfand transform image $\widehat{\mathcal{A}}$ of the Banach algebra $\mathcal{A}$, which is a uniform algebra on the spectrum of $\mathcal{A}$, we obtain an explicit description of the Shilov boundary for $\widehat{\mathcal{A}}$ for classical Banach spaces $X$ in the case where $\mathcal{A}$ is a certain Banach algebra, for instance, $\mathcal{A}_\infty (B_X)$, $\mathcal{A}_u (B_X)$ or $\mathcal{A}_{wu} (B_X)$. Some possible application of our result to the famous Corona theorem is also briefly discussed.

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arXiv2020

On quasi norm attaining operators between Banach spaces

We introduce a weakened notion of norm attainment for bounded linear operators between Banach spaces which we call \emph{quasi norm attaining operators}. An operator $T\colon X \longrightarrow Y$ between the Banach spaces $X$ and $Y$ is quasi norm attaining if there is a sequence $(x_n)$ of norm one elements in $X$ such that $(Tx_n)$ converges to some $u\in Y$ with $\|u\|=\|T\|$. Norm attaining operators in the usual sense (i.e.\ operators for which there is a point in the unit ball where the norm of its image equals the norm of the operator) and compact operators satisfy this definition. The main result of the paper is that strong Radon-Nikodým operators such as weakly compact operators can be approximated by quasi norm attaining operators (even by a stronger version of the definition), which does not hold for norm attaining operators. This allows us to give characterizations of the Radon-Nikodým property in term of the denseness of quasi norm attaining operators for both domain spaces and range spaces, extending previous results by Bourgain and Huff. We also present positive and negative results on the denseness of quasi norm attaining operators, characterize both finite dimensionality and reflexivity in terms of quasi norm attaining operators, discuss conditions to obtain that quasi norm attaining operators are actually norm attaining, study the relationship with the norm attainment of the adjoint operator, and present some stability properties. We finish the paper with some open questions.

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arXiv2019

The Bishop-Phelps-Bollobás properties in complex Hilbert spaces

In this paper we consider a stronger property than the Bishop-Phelps-Bollobás property for various classes of operators on a complex Hilbert space. The Bishop-Phelps-Bollobás {\it point} property for some class $\mathcal{A} \subset \mathcal{L}(H)$ says that if one starts with a norm one operator $T$ belonging to $\mathcal{A}$, which almost attains its norm at some norm one vector $x_0$, then there is a new operator $S$, belonging to the same class $\mathcal{A}$, which is close to $T$ and attains its norm at the same vector $x_0$. We study it for classical operators on a complex Hilbert spaces such as self-adjoint, anti-symmetric, unitary, compact, normal, and Schatten-von Neumann operators. We also solve analogous problems by replacing the norm of an operator by its numerical radius.

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arXiv2019

Emerging notions of norm attainment for Lipschitz maps between Banach spaces

We classify several notions of norm attaining Lipschitz maps which were introduced previously, and present the relations among them in order to verify proper inclusions. We also analyze some results for the sets of Lipschitz maps satisfying each of these properties to be dense or not in $\Lip(X,Y)$. For instance, we characterize a Banach space $Y$ with the Radon-Nikodým property in terms of the denseness of norm attaining Lipschitz maps with values in $Y$. Further, we introduce a property called the local directional Bishop-Phelps-Bollobás property for Lipschitz compact maps, which extends the one studied previously for scalar-valued functions, and provide some new positive results.

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arXiv2019

The Bishop-Phelps-Bollobás property and absolute sums

In this paper we study conditions assuring that the Bishop-Phelps-Bollobás property (BPBp, for short) is inherited by absolute summands of the range space or of the domain space. Concretely, given a pair (X, Y) of Banach spaces having the BPBp, (a) if Y1 is an absolute summand of Y, then (X, Y1) has the BPBp; (b) if X1 is an absolute summand of X of type 1 or \infty, then (X1, Y) has the BPBp. Besides, analogous results for the BPBp for compact operators and for the density of norm attaining operators are also given. We also show that the Bishop-Phelps-Bollobás property for numerical radius is inherited by absolute summands of type 1 or \infty. Moreover, we provide analogous results for numerical radius attaining operators and for the BPBp for numerical radius for compact operators.

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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

Bishop-Phelps-Bolloba's theorem on bounded closed convex sets

This paper deals with the \emph{Bishop-Phelps-Bollobás property} (\emph{BPBp} for short) on bounded closed convex subsets of a Banach space $X$, not just on its closed unit ball $B_X$. We firstly prove that the \emph{BPBp} holds for bounded linear functionals on arbitrary bounded closed convex subsets of a real Banach space. We show that for all finite dimensional Banach spaces $X$ and $Y$ the pair $(X,Y)$ has the \emph{BPBp} on every bounded closed convex subset $D$ of $X$, and also that for a Banach space $Y$ with property $(β)$ the pair $(X,Y)$ has the \emph{BPBp} on every bounded closed absolutely convex subset $D$ of an arbitrary Banach space $X$. For a bounded closed absorbing convex subset $D$ of $X$ with positive modulus convexity we get that the pair $(X,Y)$ has the \emph{BPBp} on $D$ for every Banach space $Y$. We further obtain that for an Asplund space $X$ and for a locally compact Hausdorff $L$, the pair $(X, C_0(L))$ has the \emph{BPBp} on every bounded closed absolutely convex subset $D$ of $X$. Finally we study the stability of the \emph{BPBp} on a bounded closed convex set for the $\ell_1$-sum or $\ell_{\infty}$-sum of a family 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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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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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.

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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.

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