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.
We introduce the so-called Bishop-Phelps-Bollobás property for positive functionals, a particular case of the Bishop-Phelps-Bollobás property for positive operators. First we show a version of the Bishop-Phelps-Bollobás theorem for positive elements and positive functionals in the dual of any Banach lattice. We also characterize the strong Bishop-Phelps-Bollobás property for positive functionals in a Banach lattice. We prove that any finite-dimensional Banach lattice has the the Bishop-Phelps-Bollobás property for positive functionals. A sufficient and a necessary condition to have the Bishop-…
We show that the set of bounded linear operators from $X$ to $X$ admits a Bishop-Phelps-Bollobás type theorem for numerical radius whenever $X$ is $\ell_1(\mathbb{C})$ or $c_0(\mathbb{C})$. As an essential tool we provide two constructive versions of the classical Bishop-Phelps-Bollobás theorem for $\ell_1(\mathbb{C})$.
We study the Bishop-Phelps-Bollobas property for numerical radius within the framework of C(K) spaces. We present several sufficient conditions on a compact space K ensuring that C(K) has the Bishop-Phelps-Bollobas property for numerical radius. In particular, we show that C(K) has such property whenever K is metrizable.
We show that the pair $(C(K),X)$ has the Bishop-Phelps-Bollobaás property for operators if $K$ is a compact Hausdorff space and $X$ is a uniformly convex space.
It is shown that the Bishop-Phelps-Bollobás theorem holds for bilinear forms on the complex $C_0(L_1)\times C_0(L_2)$ for arbitrary locally compact topological Hausdorff spaces $L_1$ and $L_2$.
Extending the celebrated result by Bishop and Phelps that the set of norm attaining functionals is always dense in the topological dual of a Banach space, Bollobás proved the nowadays known as the Bishop-Phelps-Bollobás theorem, which allows to approximate at the same time a functional and a vector in which it almost attains the norm. Very recently, two Bishop-Phelps-Bollobás moduli of a Banach space have been introduced [J. Math. Anal. Appl. 412 (2014), 697--719] to measure, for a given Banach space, what is the best possible Bishop-Phelps-Bollobás theorem in this space. In this paper we pres…
We provide a version for operators of the Bishop-Phelps-Bollobás Theorem when the domain space is the complex space $C_0(L)$. In fact we prove that the pair $(C_0(L), Y)$ satisfies the Bishop-Phelps-Bollobás property for operators for every Hausdorff locally compact space $L$ and any $\mathbb{C}$-uniformly convex space. As a consequence, this holds for $Y= L_p (μ)$ ($1 \le p < \infty $).
We study a Bishop-Phelps-Bollobás type property for Banach space operators introduced by Dantas (2017). In that paper there is a local and a global version of a natural property which is somewhat similar but simpler compared to the Bishop-Phelps-Bollobás type property for operators studied in Acosta et al. (2008). Here we characterize the mentioned local property in the setting with strictly convex domain spaces and compact operators. We show that the local property implies that the domain space has strong convexity properties.
In this paper, the notion of $c$-support points of a set in a semitopological cone is introduced. It is shown that any nonempty convex Scott closed bounded set has a $c$-support point in a cancellative $bd$-cone under certain condition. We also introduce the notion of $wd$-cone and then we prove a Bishop-Phelps type theorem for $wd$-cones, especially for normed cones, under appropriate conditions. Finally, using of the Bishop-Phelps technique, we obtain a result about the fixed points of a mapping on $s$-cones.
The Bishop-Phelps-Bollobás property for operators deals with simultaneous approximation of an operator $T$ and a vector $x$ at which $T: X\rightarrow Y$ nearly attains its norm by an operator $F$ and a vector $z$, respectively, such that $F$ attains its norm at $z$. We study the possible estimates from above and from below for parameters that measure the rate of approximation in the Bishop-Phelps-Bollobás property for operators for the case of $Y$ having the property $β$ of Lindenstrauss.
We prove that the class of positive operators from $L_\infty (μ)$ to $L_1 (ν)$ has the Bishop-Phelps-Bollobás property for any positive measures $μ$ and $ν$. The same result also holds for the pair $(c_0, \ell_1)$. We also provide an example showing that not every pair of Banach lattices satisfies the Bishop-Phelps-Bollobás property for positive operators.
We study the Bishop-Phelps-Bollobás property for operators between Banach spaces. Sufficient conditions are given for generalized direct sums of Banach spaces with respect to a~uniformly monotone Banach sequence lattice to have the approximate hyperplane series property. This result implies that Bishop-Phelps-Bollobás theorem holds for operators from $\ell_1$ into such direct sums of Banach spaces. We also show that the direct sum of two spaces with the approximate hyperplane series property has such property whenever the norm of the direct sum is absolute.
We prove that the class of Banach spaces $Y$ such that the pair $(\ell_1, Y)$ has the Bishop-Phelps-Bollobás property for operators is stable under finite products when the norm of the product is given by an absolute norm. We also provide examples showing that previous stability results obtained for that property are optimal.
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 through…
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 s…
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.
We study the Bishop-Phelps-Bollobás property for operators from $\ell_\infty ^4 $ to a Banach space. For this reason we introduce an appropiate geometric property, namely the AHSp-$\ell_\infty ^4$. We prove that spaces $Y$satisfying AHSp-$\ell_\infty ^4$ are precisely those spaces $Y$ such that $(\ell_\infty^4,Y)$ has the Bishop-Phelps-Bollobás property. We also provide classes of Banach spaces satisfying this condition. For instance, finite-dimensional spaces, uniformly convex spaces, $C_0(L)$ and $L_1 (μ)$ satisfy AHSp-$\ell_\infty ^4 $.
We prove that the class of positive operators from $L_\infty (μ)$ to $Y$ has the Bishop-Phelps-Bollobás property for any positive measure $μ$, whenever $Y$ is a uniformly monotone Banach lattice with a weak unit. The same result also holds for the pair $(c_0, Y)$ for any uniformly monotone Banach lattice $Y.$ Further we show that these results are optimal in case that $Y$ is strictly monotone.
Recently it was introduced the so-called Bishop-Phelps-Bollob{á}s property for positive operators between Banach lattices. In this paper we prove that the pair $(C_0(L), Y) $ has the Bishop-Phelps--Bollob{á}s property for positive operators, for any locally compact Hausdorff topological space $L$, whenever $Y$ is a uniformly monotone Banach lattice with a weak unit. In case that the space $C_0(L)$ is separable, the same statement holds for any uniformly monotone Banach lattice $Y .$ We also show the following partial converse of the main result. In case that $Y$ is a strictly monotone Banach l…
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.
In this paper, we introduce the notion of the Bishop-Phelps-Bollobás property for numerical radius (BPBp-$ν$) for a subclass of the space of bounded linear operators. Then, we show that certain subspaces of $\mathcal{L}(L_1(μ))$ have the BPBp-$ν$ for every finite measure $μ$. As a consequence we deduce that the subspaces of finite-rank operators, compact operators and weakly compact operators on $L_1(μ)$ have the BPBp-$ν$.
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.
In this paper we introduce the strong Bishop-Phelps-Bollobás property (sBPBp) for bounded linear operators between two Banach spaces $X$ and $Y$. This property is motivated by a Kim-Lee result which states, under our notation, that a Banach space $X$ is uniformly convex if and only if the pair $(X,\mathbb{K})$ satisfies the sBPBp. Positive results of pairs of Banach spaces $(X,Y)$ satisfying this property are given and concrete pairs of Banach spaces $(X, Y)$ failing it are exhibited. A complete characterization of the sBPBp for the pairs $(\ell_p, \ell_q)$ is also provided.
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.
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 $\e…
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 …
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 spac…
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 …
We introduce two Bishop-Phelps-Bollobás moduli which measure, for a given Banach space, what is the best possible Bishop-Phelps-Bollobás theorem in this space. We show that there is a common upper bound for these moduli for all Banach spaces and we present an example showing that this bound is sharp. We prove the continuity of these moduli and an inequality with respect to duality. We calculate the two moduli for Hilbert spaces and also present many examples for which the moduli have the maximum possible value (among them, there are $C(K)$ spaces and $L_1(μ)$ spaces). Finally, we show that if …
The Bishop-Phelps-Bollobás property deals with simultaneous approximation of an operator $T$ and a vector $x$ at which $T$ nearly attains its norm by an operator $T_0$ and a vector $x_0$, respectively, such that $T_0$ attains its norm at $x_0$. In this note we extend the already known results about {the} Bishop-Phelps-Bollobás property for Asplund operators to a wider class of Banach spaces and to a wider class of operators. Instead of proving a BPB-type theorem for each space separately we isolate two main notions: $Γ$-flat operators and Banach spaces with ACK$_ρ$ structure. In particular, we…
We study the Bishop-Phelps-Bollobás property for numerical radius restricted to the case of compact operators (BPBp-nu for compact operators in short). We show that $C_0(L)$ spaces have the BPBp-nu for compact operators for every Hausdorff topological locally compact space $L$. To this end, on the one hand, we provide some techniques allowing to pass the BPBp-nu for compact operators from subspaces to the whole space and, on the other hand, we prove some strong approximation property of $C_0(L)$ spaces and their duals. Besides, we also show that real Hilbert spaces and isometric preduals of $\…
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.
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 who…
We study the stability behavior of the Bishop-Phelps-Bollobás property for Lipschitz maps (Lip-BPB property). This property is a Lipschitz version of the classical Bishop-Phelps-Bollobás property and deals with the possibility of approximating a Lipschitz map that almost attains its (Lipschitz) norm at a pair of distinct points by a Lipschitz map attaining its norm at a pair of distinct points (relatively) very closed to the previous one. We first study the stability of this property under the (metric) sum of the domain spaces. Next, we study when it is possible to pass the Lip-BPB property fr…
The Bishop--Phelps--Bollobás Property for Extremally Disconnected Ranges: Separable and Low-Density Domains
Institute of Mathematics, Polish Academy of Sciences（波兰科学院数学研究所）；Millsaps College（米尔斯普斯学院）；School of Mathematical Sciences, Dalian University of Technology（大连理工大学数学科学学院）；Hampton University（汉普顿大学）
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…
In this paper, we introduce and study a Lipschitz version of the Bishop-Phelps-Bollobás property (Lip-BPB property). This property deals with the possibility of making a uniformly simultaneous approximation of a Lipschitz map $F$ and a pair of points at which $F$ almost attains its norm by a Lipschitz map $G$ and a pair of points such that $G$ strongly attains its norm at the new pair of points. We first show that if $M$ is a finite pointed metric space and $Y$ is a finite-dimensional Banach space, then the pair $(M,Y)$ has the Lip-BPB property, and that both finiteness assumptions are needed.…
In this paper we study a weaker form of the property $\text{\textbf{L}}_{o,o}$ called the weak $\text{\textbf{L}}_{o,o}$ and its uniform version called the weak $\text{BPB}_{\text{op}}$ which is again a weaker form the property $\text{BPB}_{\text{op}}$ for a pair of Banach spaces. We prove that a Banach space $X$ is reflexive and weakly uniformly convex if and only if the pair $(X,\mathbb{R})$ has the property weak $\text{BPB}_{\text{op}}$. We further investigate the class of all bounded linear operators from a Banach space to another Banach space satisfying the property weak $\text{\textbf{L}…
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 ab…
We introduce the notion of approximate norm attainment set of a bounded linear operator between Banach spaces and use it to obtain a complete characterization of smooth points in the space of compact linear operators, provided the domain space is reflexive and Kadets-Klee. We also apply the concept to characterize strong BPB property (sBPBp) of a pair of Banach spaces. We further introduce uniform $ ε- $BPB approximation of a bounded linear operator and uniform strong BPB property (uniform sBPBp) with respect to a given family of norm one linear operators and explore some of the relevant prope…
Bishop-Phelps Type Scalarization for Vector Optimization in Real Topological-Linear Spaces
Leibniz Universität Hannover（汉诺威莱布尼茨大学）；Martin Luther University Halle-Wittenberg（哈勒-维滕贝格马丁路德大学）；Gheorghe Asachi Technical University of Iaşi（雅西格奥尔基·阿萨奇理工大学）
We study the Bishop-Phelps-Bollobás property (BPBp for short) for compact operators. We present some abstract techniques which allows to carry the BPBp for compact operators from sequence spaces to function spaces. As main applications, we prove the following results. Let $X$, $Y$ be Banach spaces. If $(c_0,Y)$ has the BPBp for compact operators, then so do $(C_0(L),Y)$ for every locally compact Hausdorff topological space $L$ and $(X,Y)$ whenever $X^*$ is isometrically isomorphic to $\ell_1$. If $X^*$ has the Radon-Nikodým property and $(\ell_1(X),Y)$ has the BPBp for compact operators, then …
In this paper, we characterize the so called property $\textbf{L}_{o,o}$ as defined by Dantas and Rueda Zoca, for compact, weak-weak continuous bilinear maps. Motivated by this we weaken this property by defining the weak $\textbf{L}_{o,o}$ for bilinear maps. We provide equivalence of the weak $\textbf{L}_{o,o}$ property of $(X\hat{\otimes}_πY,\mathbb{R})$ for linear functionals and that of $(X,Y;\mathbb{R})$ for bilinear forms under certain conditions on $X$ and $Y$. Moreover, we have also established that under certain preassigned conditions, if a bilnear map $T$ belongs to a class which sat…
Group equivariant Radon-Nikodým property and its characterizations
Czech Technical University in Prague（布拉格捷克理工大学）；Institute of Mathematics, Czech Academy of Sciences（捷克科学院数学研究所）；Hanyang University（汉阳大学）；Faculty of Mathematics and Physics, Charles University（查理大学数学与物理学院）
In 2000 V. Lomonosov suggested a counterexample to the complex version of the Bishop-Phelps theorem on modulus support functionals. We discuss the $c_0$-analog of that example and demonstrate that the set of sup-attaining functionals is non-trivial, thus answering an open question, asked in \cite{KLMW}.
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.
We characterize real Banach spaces $Y$ such that the pair $(\ell_\infty ^n, Y)$ has the Bishop-Phelps-Bollobás property for operators. To this purpose it is essential the use of an appropriate basis of the domain space $\R^n$. As a consequence of the mentioned characterization, we provide examples of spaces $Y$ satisfying such property. For instance, finite-dimensional spaces, uniformly convex spaces, uniform algebras and $L_1(μ)$ ($μ$ a positive measure) satisfy the previous property.
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…
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 amon…
