arXiv subjects

Mingu Jung

Publications and source records attributed to Mingu Jung.

At least 37 records · Page 2Linked to original sources

arXiv2022

A note on numerical radius attaining mappings

We prove that if every bounded linear operator (or $N$-homogeneous polynomials) with the compact approximation property attains its numerical radius, then $X$ is a finite dimensional space. Moreover, we present an improvement of the polynomial James' theorem for numerical radius proved by Acosta, Becerra Guerrero and Gal$á$n in 2003. Finally, the denseness of weakly (uniformly) continuous $2$-homogeneous polynomials on a Banach space whose Aron-Berner extensions attain their numerical radii is obtained.

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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 the strong subdifferentiability of the homogeneous polynomials and (symmetric) tensor products

In this paper, we study the (uniform) strong subdifferentiability of the norms of the Banach spaces $\mathcal{P}(^N X, Y^*)$, $X \hat{\otimes}_π\cdots \hat{\otimes}_πX$ and $\hat{\otimes}_{π_s,N} X$. Among other results, we characterize when the norms of the spaces $\mathcal{P}(^N \ell_p, \ell_{q}), \mathcal{P}(^N l_{M_1}, l_{M_2})$, and $\mathcal{P}(^N d(w,p), l_{M_2})$ are strongly subdifferentiable. Analogous results for multilinear mappings are also obtained. Since strong subdifferentiability of a dual space implies reflexivity, we improve some known results on the reflexivity of spaces of $N$-homogeneous polynomials and $N$-linear mappings. Concerning the projective (symmetric) tensor norms, we provide positive results on the subsets $U$ and $U_s$ of elementary tensors on the unit spheres of $X \hat{\otimes}_π\cdots \hat{\otimes}_πX$ and $\hat{\otimes}_{π_s,N} X$, respectively. Specifically, we prove that $\hat{\otimes}_{π_s,N} \ell_2$ and $\ell_2 \hat{\otimes}_π\cdots \hat{\otimes}_π\ell_2$ are uniformly strongly subdifferentiable on $U_s$ and $U$, respectively, and that $c_0 \hat{\otimes}_{π_s} c_0$ and $c_0 \hat{\otimes}_πc_0$ are strongly subdifferentiable on $U_s$ and $U$, respectively, in the complex case.

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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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arXiv2022

Residuality in the set of norm attaining operators between Banach spaces

We study the relationship between the residuality of the set of norm attaining functionals on a Banach space and the residuality and the denseness of the set of norm attaining operators between Banach spaces. Our first main result says that if $C$ is a bounded subset of a Banach space $X$ which admit an LUR renorming satisfying that, for every Banach space $Y$, the operators $T$ from $X$ to $Y$ for which the supremum of $\|Tx\|$ with $x\in C$ is attained are dense, then the $G_δ$ set of those functionals which strongly exposes $C$ is dense in $X^*$. This extends previous results by J.\ Bourgain and K.-S.\ Lau. The particular case in which $C$ is the unit ball of $X$, in which we get that the norm of $X^*$ is Fréchet differentiable at a dense subset, improves a result by J.\ Lindenstrauss and we even present an example showing that Lindenstrauss' result was not optimal. In the reverse direction, we obtain results for the density of the $G_δ$ set of absolutely strongly exposing operators from $X$ to $Y$ by requiring that the set of strongly exposing functionals on $X$ is dense and conditions on $Y$ or $Y^*$ involving RNP and discreteness on the set of strongly exposed points of $Y$ or $Y^*$. These results include examples in which even the denseness of norm attaining operators was unknown. We also show that the residuality of the set of norm attaining operators implies the denseness of the set of absolutely strongly exposing operators provided the domain space and the dual of the range space are separable, extending a recent result for functionals. Finally, our results find important applications, among which we point out that we solve a proposed open problem showing that the unique predual of the space of Lipschitz functions from the Euclidean unit circle fails to have Lindenstrauss property A.

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arXiv2022

The Lipschitz bounded approximation property for operator ideals

In this article, we introduce the Lipschitz bounded approximation property for operator ideals. With this notion, we extend the original work done by Godefroy and Kalton and give some partial answers on the equivalence between the bounded approximation property and the Lipschitz bounded approximation property based on an arbitrary operator ideal. Furthermore, we investigate the three space problems on the preceding bounded approximation properties.

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arXiv2021

On the Lipschitz numerical index of Banach spaces

We provide some new consequences on the Lipschitz numerical radius and index which were introduced recently. More precisely, we give some renorming results on the Lipschitz numerical index, introduce a concept of Lipschitz numerical radius attaining functions in order to show that the denseness fails for an arbitrary Banach space, and study a Lipschitz version of Daugavet centers. Furthermore, we discuss the Lipschitz numerical index of vector-valued function spaces, absolute sums of Banach spaces, the Köthe-Bochner spaces, and Banach spaces which contain a dense union of increasing family of one-complemented subspaces.

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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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arXiv2021

Norm-attaining tensors and nuclear operators

Given two Banach spaces $X$ and $Y$, we introduce and study a concept of norm-attainment in the space of nuclear operators $\mathcal{N}(X,Y)$ and in the projective tensor product space $X \widehat{\otimes}_πY$. We exhibit positive and negative examples where both previous norm-attainment hold. We also study the problem of whether the class of elements which attain their norms in $\mathcal{N}(X,Y)$ and in $X\widehat{\otimes}_πY$ is dense or not. We prove that, for both concepts, the density of norm-attaining elements holds for a large class of Banach spaces $X$ and $Y$ which, in particular, covers all classical Banach spaces. Nevertheless, we present Banach spaces $X$ and $Y$ failing the approximation property in such a way that the class of elements in $X\widehat{\otimes}_πY$ which attain their projective norms is not dense. We also discuss some relations and applications of our work to the classical theory of norm-attaining operators throughout the paper.

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arXiv2021

Some remarks on the weak maximizing property

A pair of Banach spaces $(E, F)$ is said to have the weak maximizing property (WMP, for short) if for every bounded linear operator $T$ from $E$ into $F$, the existence of a non-weakly null maximizing sequence for $T$ implies that $T$ attains its norm. This property was recently introduced in an article by R. Aron, D. García, D. Pelegrino and E. Teixeira, raising several open questions. The aim of the present paper is to contribute to the better knowledge of the WMP and its limitations. Namely, we provide sufficient conditions for a pair of Banach spaces to fail the WMP and study the behaviour of this property with respect to quotients, subspaces, and direct sums, which open the gate to present several consequences. For instance, we deal with pairs of the form $(L_p[0,1], L_q[0,1])$, proving that these pairs fail the WMP whenever $p>2$ or $q<2$. We also show that, under certain conditions on $E$, the assumption that $(E, F)$ has the WMP for every Banach space $F$ implies that $E$ must be finite dimensional. On the other hand, we show that $(E, F)$ has the WMP for every reflexive space $E$ if and only if $F$ has the Schur property. We also give a complete characterization for the pairs $(\ell_s \oplus_p \ell_p, \ell_s \oplus_q \ell_q)$ to have the WMP by calculating the moduli of asymptotic uniform convexity of $\ell_s \oplus_p \ell_p$ and of asymptotic uniform smoothness of $\ell_s \oplus_q \ell_q$ when $1 < p \leq s \leq q < \infty$. We conclude the paper by discussing some variants of the WMP and presenting a list of open problems on the topic of the paper.

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arXiv2021

On norm-attainment in (symmetric) tensor products

In this paper, we introduce a concept of norm-attainment in the projective symmetric tensor product $\widehat{\otimes}_{π,s,N} X$ of a Banach space $X$, which turns out to be naturally related to the classical norm-attainment of $N$-homogeneous polynomials on $X$. Due to this relation, we can prove that there exist symmetric tensors that do not attain their norms, which allows us to study the problem of when the set of norm-attaining elements in $\widehat{\otimes}_{π,s,N} X$ is dense. We show that the set of all norm-attaining symmetric tensors is dense in $\widehat{\otimes}_{π,s,N} X$ for a large set of Banach spaces as $L_p$-spaces, isometric $L_1$-predual spaces or Banach spaces with monotone Schauder basis, among others. Next, we prove that if $X^*$ satisfies the Radon-Nikodým and the approximation property, then the set of all norm-attaining symmetric tensors in $\widehat{\otimes}_{π,s,N} X^*$ is dense. From these techniques, we can present new examples of Banach spaces $X$ and $Y$ such that the set of all norm-attaining tensors in the projective tensor product $X \widehat{\otimes}_πY$ is dense, answering positively an open question from the paper by S. Dantas, M. Jung, Ó. Roldán and A. Rueda Zoca.

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arXiv2021

On the existence of non-norm-attaining operators

In this paper we provide necessary and sufficient conditions for the existence of non-norm-attaining operators in $\mathcal{L}(E, F)$. By using a theorem due to Pfitzner on James boundaries, we show that if there exists a relatively compact set $K$ of $\mathcal{L}(E, F)$ (in the weak operator topology) such that $0$ is an element of its closure (in the weak operator topology) but it is not in its norm closed convex hull, then we can guarantee the existence of an operator which does not attain its norm. This allows us to provide the following generalization of results due to Holub and Mujica. If $E$ is a reflexive space, $F$ is an arbitrary Banach space, and the pair $(E, F)$ has the bounded compact approximation property, then the following are equivalent: (i) $\mathcal{K}(E, F) = \mathcal{L}(E, F)$; (ii) Every operator from $E$ into $F$ attains its norm; (iii) $(\mathcal{L}(E,F), τ_c)^* = (\mathcal{L}(E, F), \| \cdot \|)^*$; where $τ_c$ denotes the topology of compact convergence. We conclude the paper presenting a characterization of the Schur property in terms of norm-attaining operators.

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arXiv2021

Daugavet points in projective tensor products

In this paper, we are interested in studying when an element $z$ in the projective tensor product $X \widehat{\otimes}_πY$ turns out to be a Daugavet point. We prove first that, under some hypothesis, the assumption of $X \widehat{\otimes}_πY$ having the Daugavet property implies the existence of a great amount of isometries from $Y$ into $X^*$. Having this in mind, we provide methods for constructing non-trivial Daugavet points in $X \widehat{\otimes}_πY$. We show that $C(K)$-spaces are examples of Banach spaces such that the set of the Daugavet points in $C(K) \widehat{\otimes}_πY$ is weakly dense when $Y$ is a subspace of $C(K)^*$. Finally, we present some natural results on when an elementary tensor $x \otimes y$ is a Daugavet point.

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arXiv2021

Daugavet points and $Δ$-points in Lipschitz-free spaces

We study Daugavet points and $Δ$-points in Lipschitz-free Banach spaces. We prove that, if $M$ is a compact metric space, then $μ\in S_{\mathcal F(M)}$ is a Daugavet point if, and only if, there is no denting point of $B_{\mathcal F(M)}$ at distance strictly smaller than two from $μ$. Moreover, we prove that if $x$ and $y$ are connectable by rectifiable curves of lenght as close to $d(x,y)$ as we wish, then the molecule $m_{x,y}$ is a $Δ$-point. Some conditions on $M$ which guarantee that the previous implication reverses are also obtained. As a consequence of our work, we show that Lipschitz-free spaces are natural examples of Banach spaces where we can guarantee the existence of $Δ$-points which are not Daugavet points.

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arXiv2020

Norm attaining operators which satisfy a Bollobás type theorem

In this paper, we are interested in studying the set $\mathcal{A}_{\|\cdot\|}(X, Y)$ of all norm-attaining operators $T$ from $X$ into $Y$ satisfying the following: given $ε>0$, there exists $η$ such that if $\|Tx\| > 1 - η$, then there is $x_0$ such that $\| x_0 - x\| < ε$ and $T$ itself attains its norm at $x_0$. We show that every norm one functional on $c_0$ which attains its norm belongs to $\mathcal{A}_{\|\cdot\|}(c_0, \mathbb{K})$. Also, we prove that the analogous result holds neither for $\mathcal{A}_{\|\cdot\|}(\ell_1, \mathbb{K})$ nor $\mathcal{A}_{\|\cdot\|}(\ell_{\infty}, \mathbb{K})$. Under some assumptions, we show that the sphere of the compact operators belongs to $\mathcal{A}_{\|\cdot\|}(X, Y)$ and that this is no longer true when some of these hypotheses are dropped. The analogous set $\mathcal{A}_{nu}(X)$ for numerical radius of an operator instead of its norm is also defined and studied. We present a complete characterization for the diagonal operators which belong to the sets $\mathcal{A}_{\| \cdot \|}(X, X)$ and $\mathcal{A}_{nu}(X)$ when $X=c_0$ or $\ell_p$. As a consequence, we get that the canonical projections $P_N$ on these spaces belong to our sets. We give examples of operators on infinite dimensional Banach spaces which belong to $\mathcal{A}_{\| \cdot \|}(X, X)$ but not to $\mathcal{A}_{nu}(X)$ and vice-versa. Finally, we establish some techniques which allow us to connect both sets by using direct sums.

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arXiv2020

Group invariant separating polynomials on a Banach space

We study the group invariant continuous polynomials on a Banach space $X$ that separate a given set $K$ in $X$ and a point $z$ outside $K$. We show that if $X$ is a real Banach space, $G$ is a compact group of $\mathcal{L} (X)$, $K$ is a $G$-invariant set in $X$, and $z$ is a point outside $K$ that can be separated from $K$ by a continuous polynomial $Q$, then $z$ can also be separated from $K$ by a $G$-invariant continuous polynomial $P$. It turns out that this result does not hold when $X$ is a complex Banach space, so we present some additional conditions to get analogous results for the complex case. We also obtain separation theorems under the assumption that $X$ has a Schauder basis which give applications to several classical groups. In this case, we obtain characterizations of points which can be separated by a group invariant polynomial from the closed unit ball.

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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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