On some local Bishop-Phelps-Bollob\'as properties

TL;DR

This paper investigates specific local Bishop-Phelps-Bollobás properties for bounded linear operators, analyzing their relationships with geometric space properties and providing a comparative diagram of these properties.

Contribution

It introduces and studies two new local Bishop-Phelps-Bollobás properties, L_{p, o} and L_{o, p}, and explores their connections with geometric properties of Banach spaces.

Findings

Defined and analyzed the properties L_{p, o} and L_{o, p}

Established relations between these properties and space geometries like strict convexity and local uniform rotundity

Provided a diagram comparing all known Bishop-Phelps-Bollobás type properties

Abstract

We continue a line of study about some local versions of Bishop-Phelps-Bollob\'as 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 $\beta$ of Lindenstrauss. At the end of the paper, we present a diagram comparing all the existing Bishop-Phelps-Bollob\'as type properties with each other. Some open questions are left throughout the article.

Equations

$$\Vert T\Vert =\underset{x\in S_X}{\sup }\Vert T(x)\Vert =\Vert T(x_0)\Vert .$$

$$\Vert T(x)\Vert >1-\eta (\epsilon ),$$

$$\Vert S(x_0)\Vert =1,\Vert x_0-x\Vert <\epsilon ,\text{and}\Vert S-T\Vert <\epsilon .$$

$$\Vert T(x_1)\Vert =1\text{and}\Vert x_0-x_1\Vert <\epsilon .$$

$$\Vert T(x)\Vert >1-\eta (\epsilon ,T),$$

$$\Vert S(x)\Vert =1\text{mbox}and\Vert S-T\Vert <\epsilon .$$

$$\Vert T(x)\Vert >1-\eta (\epsilon ,x),$$

$$\Vert T(x_0)\Vert =1\text{mbox}and\Vert x_0-x\Vert <\epsilon .$$

$$\text{if X is reflexive,}\text{X is LUR}\Rightarrow \text{ (X,K) has the Lo,p}\Rightarrow \text{X is strictly convex}.$$

$$\underset{n}{\lim }\Vert x_n+x\Vert =2⟹\underset{n}{\lim }\Vert x_n-x\Vert =0.$$

$$1⩾\left|x_n^{\ast }(x_0)\right|⩾1-\frac{1}{n}$$

$$1⩾\Vert \frac{c_n{x}_n+x_0}{2}\Vert ⩾\left|\frac{x_n^{\ast }(c_n{x}_n)+x_n^{\ast }(x_0)}{2}\right|⟶1.$$

$$\text{Re}{x}^{\ast }(x)⩽1-\min \left\{\eta \left(\frac{{\epsilon }^2}{64},x\right),\eta \left(\frac{{\epsilon }^2}{64},y\right),\frac{{\epsilon }^2}{64}\right\}$$

$$\text{Re}{x}^{\ast }(y)⩽1-\min \left\{\eta \left(\frac{{\epsilon }^2}{64},x\right),\eta \left(\frac{{\epsilon }^2}{64},y\right),\frac{{\epsilon }^2}{64}\right\}.$$

$$\frac{\Vert x+y\Vert }{2}$$

$$T_n(x):=\left(1-\frac{1}{n}\right)v_1^{\ast }(x)v_1+v_2^{\ast }(x)v_2\left(x\in X\right).$$

$$\Vert T_n(x)\Vert =\Vert \left(1-\frac{1}{n}\right)v_1^{\ast }(x)v_1+v_2^{\ast }(x)v_2\Vert$$

$$\Vert T_n(v_1)\Vert =1-\frac{1}{n}>1-\eta ({\epsilon }_0,v_1),$$

$$1=\Vert T_n(x_0)\Vert ⩽\left(1-\frac{1}{n}\right)\Vert x_0\Vert +\frac{1}{n}|v_2^{\ast }(x_0)|⩽1$$

$$T_n(x):=\left(\left(1-\frac{1}{n}\right)x_1^{\ast }(x),x_2^{\ast }(x)\right)(x\in X).$$

$$\Vert T_n(x_1){\Vert }_{\infty }=1-\frac{1}{n}>1-\eta ({\epsilon }_0,x_1),$$

$$1=\Vert T_n(z){\Vert }_{\infty }=\max \left\{\left(1-\frac{1}{n}\right)|x_1^{\ast }(z)|,|x_2^{\ast }(z)|\right\},$$

$$\psi (\epsilon ,T)=\underset{i\in I}{\min }\{\eta (\epsilon ,y_i^{\ast }\circ T)\}>0.$$

$$1+\rho \left(\frac{{\epsilon }_0}{4}+\xi \right)<\left(1+\frac{{\epsilon }_0}{4}\right)(1-\xi ).$$

$$y_k^{\ast }(T_0(x_0))>1-\psi (\xi ,T_0)⩾1-\eta (\xi ,y_k\circ T_0).$$

$$U(x):=T_0(x)+\left[\left(1+\frac{{\epsilon }_0}{4}\right)x_1^{\ast }(x)-y_k^{\ast }\circ T_0^{\ast }(x)\right]y_k(x\in X).$$

$$\Vert y_j^{\ast }\circ U\Vert ⩽1+\rho \left(\frac{{\epsilon }_0}{4}+\xi \right)<\left(1+\frac{{\epsilon }_0}{4}\right)(1-\xi )<1+\frac{{\epsilon }_0}{4}\text{mbox}and\Vert y_k^{\ast }\circ U\Vert =1+\frac{{\epsilon }_0}{4}.$$

$$\Vert T(E_i{z}_i)\Vert ⟶1\text{mbox}and\Vert P_{j}T-c{\tilde{E}}_i{z}_i^{\ast }\Vert >\frac{1}{2^{1-\frac{1}{i}}}$$

$$\Vert S(x_0)\Vert =1,\Vert x_0-x\Vert <\epsilon ,\text{mbox}and\Vert S-T\Vert <\epsilon .$$

Full text

The second author is the corresponding author. The first author was supported by the project OPVVV CAAS CZ.02.1.01/0.0/0.0/16_019/0000778. The second author was partially supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (NRF-2017R1C1B1002928). The third author was supported by the Research program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2016R1D1A1B03934771). The fourth author was partially supported by CONICET PIP 11220130100329CO

1. Introduction

One of the main results in the theory of norm attaining functions defined on Banach spaces was proved by Errett Bishop and Robert R. Phelps in [7]. They showed that the set of all functionals which attain the maximum on a nonempty closed bounded convex subset $S$ of a real Banach space $X$ is norm dense in the dual space $X^{*}$. On the other hand, Victor Lomonosov presented in [19] an example which shows that this statement cannot be extended to the complex case by constructing a closed bounded convex subset of some Banach space with no support points. Here, we are interested to study this result when $S$ is the closed unit ball, which simply says that the set of all norm attaining functionals defined on a real or complex Banach space $X$ is dense in $X^{*}$ (see also [6]). We will refer this last statement as the Bishop-Phelps theorem. Joram Lindenstrauss was the first mathematician who considered the vector valued case of the Bishop-Phelps theorem (see [18]). He presented a counterexample which proves that this theorem is no longer valid for bounded linear operators in general. Nevertheless, he gave some necessary conditions to get a Bishop-Phelps type theorem for this class of functions. For instance, if the domain $X$ is a reflexive Banach space, then it is true that the set of all norm attaining operators from $X$ into any Banach space $Y$ is dense in the set of all operators from $X$ into $Y$. After Lindenstrauss, a lot of attention has been paid on this topic. We refer to the survey paper [1] and the references therein for more information about denseness of norm attaining functions in various directions.

In [8], Béla Bollobás proved a stronger version of the Bishop-Phelps theorem, in such a way that whenever a norm-one functional $x^{*}$ almost attains its norm at some norm-one point $x$, it is possible to find a new norm-one functional $y^{*}$ and a new norm-one point $y$ such that $y^{*}$ attains its norm at $y$, $y$ is close to $x$, and $y^{*}$ is close to $x^{*}$. Since the norm of a functional is defined as a supremum and we can always take some point such that a given functional almost attains its norm, Bollobás result says that in the Bishop-Phelps theorem one can control the distances between the involved points and functionals. This result is known nowadays as the Bishop-Phelps-Bollobás theorem. Motivated by Lindenstrauss work, in 2008, María Acosta, Richard Aron, Domingo García, and Manuel Maestre initiated the study of the Bishop-Phelps-Bollobás theorem in the vector-valued case (see [3]). They found conditions on Banach spaces $X$ and $Y$ in order to get a Bishop-Phelps-Bollobás type theorem for operators from $X$ into $Y$. For instance, they characterized those spaces $Y$ such that the Bishop-Phelps-Bollobás theorem holds for operators from $\ell_{1}$ into $Y$. After more than 10 years of [3], there is a huge literature about this topic and we refer the reader to [2, 4, 5, 10, 11, 17] and the references therein for further information. Many different variants of the Bishop-Phelps-Bollobás theorem were introduced during the last years. For some of them, we refer the recent papers [12, 13, 14, 15]. Our aim is to study local versions of these properties, as in [16]. Before we explain exactly what this means, let us introduce some notation and necessary preliminaries.

We work on Banach spaces over the field $\mathbb{K}$, which can be the real or complex numbers. We denote by $S_{X}$, $B_{X}$, and $X^{*}$ the unit sphere, the unit ball, and the topological dual of $X$, respectively. The symbol $\mathcal{L}(X,Y)$ stands for the set of all bounded linear operators from $X$ into $Y$ and we say that $T\in\mathcal{L}(X,Y)$ attains its norm (or it is norm attaining) if there is $x_{0}\in S_{X}$ such that

[TABLE]

Following [3], we say that a pair of Banach spaces $(X,Y)$ satisfies the Bishop-Phelps-Bollobás property (BPBp, for short) if given $\varepsilon>0$, there is $\eta(\varepsilon)>0$ such that whenever $T\in\mathcal{L}(X,Y)$ with $\|T\|=1$ and $x\in S_{X}$ are such that

[TABLE]

there are $S\in\mathcal{L}(X,Y)$ with $\|S\|=1$ and $x_{0}\in S_{X}$ such that

[TABLE]

When $x_{0}=x$ in the previous definition, we say that $(X,Y)$ has the Bishop-Phelps-Bollobás point property (BPBpp, for short); this property was defined and studied in [13, 14]. If instead of fixing the point $x$ (as in the BPBpp) we fix the operator $T$, we say that $(X,Y)$ has the Bishop-Phelps-Bollobás operator property (see [12, 15]). That is, $(X,Y)$ has the Bishop-Phelps-Bollobás operator property (BPBop, for short) if given $\varepsilon>0$, there is $\eta(\varepsilon)>0$ such that whenever $T\in\mathcal{L}(X,Y)$ with $\|T\|=1$ and $x_{0}\in S_{X}$ are such that $\|T(x_{0})\|>1-\eta(\varepsilon)$, there is $x_{1}\in S_{X}$ such that

[TABLE]

Notice that the BPBp, BPBpp, and BPBop are uniform properties in the sense that $\eta$ depends just on a given $\varepsilon>0$. As we already mentioned before, we are interested to study the situations when $\eta$ depends not only $\varepsilon$, but also on the vector $x$ or the operator $T$. Some of them were already studied by the authors of the present paper in [16] and here we are using a similar notation. We state now the definition of the two local properties on which we will focus.

Definition 1.1**.**

(a) A pair $(X,Y)$ has the Lp,o if given $\varepsilon>0$ and $T\in\mathcal{L}(X,Y)$ with $\|T\|=1$, then there is $\eta(\varepsilon,T)>0$ such that whenever $x\in S_{X}$ satisfies

[TABLE]

there is $S\in\mathcal{L}(X,Y)$ with $\|S\|=1$ such that

[TABLE]

(b) A pair $(X,Y)$ has the Lo,p if given $\varepsilon>0$ and $x\in S_{X}$, then there is $\eta(\varepsilon,x)>0$ such that whenever $T\in\mathcal{L}(X,Y)$ with $\|T\|=1$ satisfies

[TABLE]

there is $x_{0}\in S_{X}$ such that

[TABLE]

Let us clarify the notation: in the symbol L□,△, both $\square$ and $\triangle$ can be $p$ or $o$, which are the initials of the words point and operator, respectively. If $(X,Y)$ has the L□,△, then it means that we fix $\square$ and $\eta$ depends on $\triangle$. So, for instance, in Lp,o we fix a norm-one point $x$ and $\eta$ depends on a norm-one operator $T$. In [16], properties Lp,p and Lo,o were addressed. Both of them are deeply related to geometric properties of the involved Banach spaces as local uniform rotundity or some of the Kadec-Klee properties. In fact, it turns out that the Lp,p for linear functionals defined on a Banach space $X$ is equivalent to the strong subdifferentiability of the norm of $X$ (see [16, Theorem 2.3]). It is also a straightforward observation that if $X$ is reflexive then Lo,o is dual to Lp,p in the sense that $(X,\mathbb{K})$ has the Lo,o if and only if $(X^{*},\mathbb{K})$ has the Lp,p (see [16, Proposition 2.2]). Here, we are interested to give continuity in the study of these type of properties, but now investigating Lp,o and Lo,p.

We describe now the contents of this paper. In first place, we obtain sufficient and necessary conditions for a pair $(X,\mathbb{K})$ to have the Lo,p, in terms of some rotundity properties of $X$. Specifically, we prove that

[TABLE]

We also prove that there exists a dual relation between properties Lp,o and Lo,p in the functional case and, as a consequence, we get that if $X$ is reflexive and $X^{*}$ is locally uniformly rotund, then the pair $(X,\mathbb{K})$ satisfies the Lp,o. As a consequence of (1) and the dual relation between Lp,o and Lo,p we see that, even for $2$-dimensional spaces, there is a Banach space $X$ such that the pair $(X,\mathbb{K})$ fails both properties. This establish a difference between the local properties Lp,o, Lo,p and Lp,p, Lo,o, since the latter properties hold in the finite-dimensional case. Concerning linear operators, we show that pairs of the form $(X,\ell_{\infty}^{2})$ and $(Z,Z)$, where $\dim(X)\geqslant 2$ and $Z$ is a $2$-dimensional space, fail property Lo,p. The situation with pairs like $(X,\ell_{\infty}^{2})$ changes for property Lp,o: we prove that if $Y$ has property $\beta$ of Lindenstrauss with a finite index set $I$, then the pair $(X,Y)$ satisfies the Lp,o whenever $(X,\mathbb{K})$ does. Nevertheless, this is no long true when $I$ is infinite and we present a counterexample to prove this. Finally, we show that $(\ell_{1},Y)$ and $(c_{0},Y)$ fail both properties for all Banach spaces $Y$. In the last part of the paper, we compare all these properties with each other and also with the BPBp, BPBpp, and BPBop.

2. The results

In this section, we show the results we have for both properties Lo,p and Lp,o. We start by proving some positive results. Notice that it is clear that the BPBpp implies the Lp,o. Hence, there are some immediate examples of pairs of Banach spaces $(X,Y)$ satisfying the Lp,o (see [13, 14] for positive results on the BPBpp). It is also clear that the BPBop implies the Lo,p, although this does not provide many examples, since the BPBop holds only for the pairs $(\mathbb{K},Y)$ for every Banach space $Y$ and $(X,\mathbb{K})$ for uniformly convex Banach spaces $X$ (see [14, 17]). Here, we get other examples of pairs $(X,Y)$ satisfying the properties Lp,o and Lo,p (see Proposition 2.1, Corollary 2.2 and Theorems 2.8 and 2.10). Recall that a Banach space is strictly convex if $\|\frac{x+y}{2}\|<1$ whenever $x,y\in S_{X}$, $x\neq y$, and that is locally uniformly rotund (LUR, for short) if for all $x,x_{n}\in S_{X}$,

[TABLE]

It is a well-known fact that if $X$ is LUR, then is strictly convex.

Proposition 2.1**.**

Let $X$ be a Banach space.

1. (i)

If $X$ is reflexive and LUR, then the pair $(X,\mathbb{K})$ has the Lo,p. 2. (ii)

If $X$ has the Radon-Nikodým property and $(X,\mathbb{K})$ has the Lo,p, then $X$ is strictly convex.

Proof.

(i). Otherwise, there are $\varepsilon_{0}>0$ and $x_{0}\in S_{X}$ such that for every $n\in\mathbb{N}$, there is $x_{n}^{*}\in S_{X^{*}}$ with

[TABLE]

such that whenever $x\in S_{X}$ satisfies $\|x-x_{0}\|<\varepsilon_{0}$, we have that $|x_{n}^{*}(x)|<1$. Since $X$ is reflexive, there is $x_{n}\in S_{X}$ such that $|x_{n}^{*}(x_{n})|=1$ for every $n\in\mathbb{N}$. For suitable modulus 1 constants $c_{n}$, we have that

[TABLE]

Since $X$ is LUR, we see that $\|c_{n}x_{n}-x_{0}\|\longrightarrow 0$ as $n\rightarrow\infty$. Then, we must have $|x_{n}^{*}(c_{n}x_{n})|<1$ for large enough $n$ and this is a contradiction.

(ii). Let $\varepsilon>0$ and $x,y\in S_{X}$ such that $\|x-y\|\geqslant\varepsilon$. We want to show that there is $\delta(\varepsilon,x,y)>0$ such that $\frac{\|x+y\|}{2}\leqslant 1-\delta(\varepsilon,x,y)$. Let $\Gamma$ be the set of all bounded linear functionals in $S_{X^{*}}$ that strongly expose $B_{X^{*}}$. Following the proof of [17, Theorem 2.1] (with slight modifications) we get that each $x^{*}\in\Gamma$ satisfies either

[TABLE]

or

[TABLE]

where $\eta(\cdot,\cdot)$ is the function in the definition of Lo,p. Now, since $X$ has the Radon-Nikodým property we have that $\Gamma$ is dense in $S_{X^{*}}$ (see [9, 21]) and, consequently,

[TABLE]

Then, $\delta(\varepsilon,x,y)=\frac{1}{2}\min\left\{\eta\left(\frac{\varepsilon^{2}}{64},x\right),\eta\left(\frac{\varepsilon^{2}}{64},y\right),\frac{\varepsilon^{2}}{64}\right\}$. ∎

We do not know if reflexivity (or the Radon-Nikodým property) is a necessary condition for the Lo,p in the above proposition. However, if we assume that $X$ is reflexive, we have the following consequence.

Corollary 2.2**.**

Let $X$ be a reflexive Banach space.

1. (i)

If $X^{*}$ is LUR, then the pair $(X,\mathbb{K})$ has the Lp,o. 2. (ii)

If $(X,\mathbb{K})$ has the Lp,o, then $X^{*}$ is strictly convex.

Notice that (ii) is just a consequence of Proposition 2.1.(ii). To see (i), we prove dual relations between the properties Lp,o and Lo,p for functionals. Corollary 2.2.(i) will then follow as a combination of Propositions 2.1.(i) and 2.3.

Proposition 2.3**.**

Let $X$ be a Banach space. If $(X^{*},\mathbb{K})$ has the Lo,p, then $(X,\mathbb{K})$ has the Lp,o.

Proof.

Assume $\varepsilon>0$ and $x^{*}\in X^{*}$ with $\|x^{*}\|=1$ are given. By hypothesis, we can take the constant $\eta(\varepsilon,x^{*})>0$ for the Lo,p of the pair $(X^{*},\mathbb{K})$. Let $x\in S_{X}$ be such that $|x^{*}(x)|>1-\eta(\varepsilon,x^{*})$. Using the canonical inclusion $\hat{\leavevmode\nobreak\ }:X\longrightarrow X^{**}$, we have $|\hat{x}(x^{*})|=|x^{*}(x)|>1-\eta(\varepsilon,x^{*})$, and so there exists $x_{1}^{*}\in S_{X^{*}}$ such that $|\hat{x}(x_{1}^{*})|=|x_{1}^{*}(x)|=1$ and $\|x_{1}^{*}-x^{*}\|<\varepsilon$. This proves that $(X,\mathbb{K})$ has the Lp,o. ∎

Proposition 2.4**.**

Let $X$ be a reflexive Banach space. The pair $(X,\mathbb{K})$ has the Lp,o if and only if $(X^{*},\mathbb{K})$ has the Lo,p.

Proof.

From Proposition 2.3, we need to prove just the ‘only if’ part. Assume $\varepsilon>0$ and $x^{*}\in S_{X^{*}}$ are given. By hypothesis, there is the constant $\eta(\varepsilon,x^{*})>0$ for the Lp,o of the pair $(X,\mathbb{K})$. Let $x^{**}\in X^{**}$ with $\|x^{**}\|=1$ be such that $|x^{**}(x^{*})|>1-\eta(\varepsilon,x^{*})$. Using the canonical inclusion $\hat{\leavevmode\nobreak\ }:X\longrightarrow X^{**}$ and the reflexivity of $X$, there exists $x\in X$ such that $\hat{x}=x^{**}$. Hence, we have $|x^{**}(x^{*})|=|x^{*}(x)|>1-\eta(\varepsilon,x^{*})$, and so there exists $z\in S_{X}$ such that $|x^{*}(z)|=1$ and $\|z-x\|<\varepsilon$. The bidual element $\hat{z}$ is the desired one for the Lo,p of the pair $(X^{*},\mathbb{K})$. ∎

At this point we would like to stress some open problems that we are not able to solve. The first one was mentioned above. The second one relies on the fact that, those spaces $X$ for which we can assure that $(X,\mathbb{K})$ has the Lo,p (respectively, Lp,o), satisfy also that $(X,\mathbb{K})$ has the Lo,o (respectively, Lp,p). Indeed, it was already observed (see the discussion above [16, Theorem 2.5]) that if $X$ is reflexive and LUR, then $(X,\mathbb{K})$ has the Lo,o.

Question 1: Does property Lo,p (or Lp,o) of the pair $(X,\mathbb{K})$ imply reflexivity of $X$?

Question 2: Does property Lo,p (respectively, Lp,o) imply property Lo,o (respectively, Lp,p)?

It is known that, for finite-dimensional Banach spaces $X$ and $Y$, the pair $(X,Y)$ satisfies property Lp,p ([16, Proposition 2.9]). Besides this, it was proved in [12] that if $X$ is finite dimensional, then the pair $(X,Y)$ has the Lo,o for every Banach space $Y$. However, this is not the case for both properties Lp,o and Lo,p even for linear functionals defined on 2-dimensional spaces. This is an immediate consequence of Proposition 2.4 and item (ii) in Proposition 2.1. In what follows, we denote by $\ell_{p}^{n}$ the $n$-dimensional space endowed with the $p$-norm and $(e_{i})_{i=1}^{n}$ their canonical basis.

Proposition 2.5**.**

The pairs $(\ell_{1}^{n},\mathbb{K})$ and $(\ell_{\infty}^{n},\mathbb{K})$ fail both Lp,o and Lo,p for $n\geqslant 2$.

Next result shows that all the pairs of the form $(X,X)$, for 2-dimensional Banach spaces $X$ fails the Lo,p for linear operators.

Proposition 2.6**.**

Let $X$ be a $2$-dimensional Banach space. Then, the pair $(X,X)$ fails the Lo,p.

Proof.

Consider $\{(v_{1},v^{*}_{1}),(v_{2},v^{*}_{2})\}$ the Auerbach basis of the space $X$. Then, for every $x\in X$, we have that $x=v_{1}^{*}(x)v_{1}+v_{2}^{*}(x)v_{2}$. Let us suppose by contradiction that the pair $(X,X)$ satisfies the Lo,p with some function $\eta(\cdot,\cdot)$ and let $n\in\mathbb{N}$ be such that $\frac{1}{n}<\eta(\varepsilon_{0},v_{1})$ for a fixed positive number $\varepsilon_{0}\in(0,1)$. Define $T_{n}:X\longrightarrow X$ by

[TABLE]

We see that

[TABLE]

for arbitrary $x\in B_{X}$. This implies that $\|T_{n}\|=1=\|T_{n}(v_{2})\|$. Now, since

[TABLE]

there is $x_{0}\in S_{X}$ such that $\|T_{n}(x_{0})\|=1$ and $\|x_{0}-v_{1}\|<\varepsilon_{0}$. On the other hand, we have that

[TABLE]

which implies $|v_{2}^{*}(x_{0})|=1$. This gives us a contradiction since $1>\varepsilon_{0}>\|x_{0}-v_{1}\|>|v_{2}^{*}(x_{0})|$. ∎

We get another negative result for the property Lo,p when the range space is $\ell_{\infty}^{2}$.

Proposition 2.7**.**

Let $X$ be a Banach space with $\dim(X)\geqslant 2$. Then, $(X,\ell_{\infty}^{2})$ fails the Lo,p.

Proof.

Let $x_{1},x_{2}\in S_{X}$ and $x_{1}^{*},x_{2}^{*}\in S_{X^{*}}$ be such that $x_{i}^{*}(x_{j})=\delta_{ij}$ for $i,j=1,2$ (we may choose such elements by taking the Hahn-Banach extension of functionals of the Auerbach basis on a 2-dimensional subspace of $X$). We assume that the pair $(X,\ell_{\infty}^{2})$ has the Lo,p with some function $\eta(\cdot,\cdot)$ and consider $n\in\mathbb{N}$ such that $\frac{1}{n}<\eta(\varepsilon_{0},x_{1})$ for a fixed positive number $\varepsilon_{0}\in(0,1)$. Define $T_{n}:X\longrightarrow\ell_{\infty}^{2}$ by

[TABLE]

Then $\|T_{n}\|\leqslant 1$ and $\|T_{n}(x_{2})\|_{\infty}=1$, which implies $\|T_{n}\|=1$. Since

[TABLE]

there is $z\in S_{X}$ such that $\|T_{n}(z)\|_{\infty}=1$ and $\|z-x_{1}\|<\varepsilon_{0}$. So, since

[TABLE]

we have that $|x_{2}^{*}(z)|=1$. Nevertheless, we have that $1>\varepsilon_{0}>\|z-x_{1}\|\geqslant|x_{2}^{*}(z)-x_{2}^{*}(x_{1})|=|x_{2}^{*}(z)|$, which gives a contradiction. ∎

Taking into account Propositions 2.6 and 2.7 and Corollary 2.12 below, we leave the following open question.

Question 3: Are there spaces $X,Y$ with dim$(X)$, dim$(Y)$ $\geqslant 2$ such that $(X,Y)$ satisfies property Lo,p?

Although we have a negative result in Proposition 2.7 for the Lo,p, the situation with property Lp,o is quite different. Indeed, we will prove that when we assume that the pair $(X,\mathbb{K})$ has the Lp,o, so does the pair $(X,\ell_{\infty}^{2})$. In fact, we get a more general result for Banach spaces satisfying property $\beta$ of Lindenstrauss (see [18]). We say that a Banach space $Y$ has property $\beta$ with a index set $I$ and a constant $0\leqslant\rho<1$ if there is a set $\{(y_{i},y_{i}^{*}):i\in I\}\subset S_{Y}\times S_{Y^{*}}$ such that

• •

$y_{i}^{*}(y_{i})=1$ for all $i\in I$,

• •

$|y_{i}^{*}(y_{j})|\leqslant\rho<1$ for all $i,j\in I$ with $i\not=j$, and

• •

$\|y\|=\sup_{i\in I}|y_{i}^{*}(y)|$ for all $y\in Y$.

Examples of Banach spaces satisfying such a property are $c_{0}(I)$ and $\ell_{\infty}(I)$ by taking $\{(e_{i},e_{i}^{*}):i\in I\}$, the canonical biorthogonal system of these spaces. For the next result, we notice that in Definition 1.1.(a) one can use $T\in B_{\mathcal{L}(X,Y)}$ instead of $T\in S_{\mathcal{L}(X,Y)}$ by a simple change of parameters.

Theorem 2.8**.**

Let $X,Y$ be Banach spaces. Suppose that $(X,\mathbb{K})$ satisfies the Lp,o and assume that $Y$ has property $\beta$ with a finite index set $I$ and constant $\rho$. Then, the pair $(X,Y)$ has the Lp,o.

Proof.

The proof is similar to [13, Proposition 2.4], but we give the details for sake of completeness. Let $I$ be a finite set and $\{(y_{i},y_{i}^{*}):i\in I\}\subset S_{Y}\times S_{Y^{*}}$ be the set of property $\beta$. Consider $\eta(\cdot,\cdot)$, the function for the pair $(X,\mathbb{K})$, which satisfies the Lp,o. For each $\varepsilon>0$ and $T\in S_{\mathcal{L}(X,Y)}$, we define

[TABLE]

Fixed $\varepsilon_{0}>0$ and $T_{0}\in S_{\mathcal{L}(X,Y)}$, we choose $0<\xi<\frac{\varepsilon_{0}}{4}$ such that

[TABLE]

Now, let $x_{0}\in S_{X}$ be such that $\|T_{0}(x_{0})\|>1-\psi(\xi,T_{0})$. By the definition of property $\beta$ and the construction of $\psi$, there exists $k\in I$ such that

[TABLE]

Hence, there exists a functional $x_{1}^{*}\in S_{X^{*}}$ such that $|x_{1}^{*}(x_{0})|=1$ and $\|x_{1}^{*}-y_{k}^{*}\circ T_{0}^{*}\|<\xi$. Now, we define $U:X\longrightarrow Y$ by

[TABLE]

We have that $\|U-T_{0}\|<\frac{\varepsilon_{0}}{4}+\xi<\frac{\varepsilon_{0}}{2}$. Moreover, for arbitrary $j\neq k$, we have that

[TABLE]

Then, $U$ attains its norm at $x_{0}$ and so the operator $V:=U/\|U\|$ is the one we were looking for. ∎

The main difference between [13, Proposition 2.4] and Theorem 2.8 is the cardinality of the index set $I$. Indeed, in [13, Proposition 2.4], we see that the set $I$ does not need to be finite, since if $X$ is uniformly smooth, then the pair $(X,\mathbb{K})$ satisfies the BPBpp and so does the Lp,o, which is, in this case, uniform, in the sense that $\eta$ depends only on a given $\varepsilon>0$. This gives that $\psi(\varepsilon,T)=\inf_{i\in I}\{\eta(\varepsilon,y_{i}^{*}\circ T)\}$, in the proof of Theorem 2.8, is strictly bigger than [math]. Naturally, one may ask whether the same result holds for infinite index sets. It turns out that this is not the case. To see why this happens, we consider the Banach space $X=\left[\oplus_{i=2}^{\infty}\ell_{i}^{2}\right]_{\ell_{2}}$, the $\ell_{2}$ direct sum of 2-dimensional $\ell_{i}$-spaces. We have that $X^{*}$ is a reflexive LUR Banach space (see, for example, [20, Theorem 1.1]). Hence, the pair $\left(\left[\oplus_{i=2}^{\infty}\ell_{i}^{2}\right]_{\ell_{2}},\mathbb{K}\right)$ satisfies property Lp,o by Corollary 2.2. Recall that $\ell_{\infty}$ satisfies property $\beta$ with $I=\mathbb{N}$ and $\rho=0$. Our counterexample is described in the next proposition.

Proposition 2.9**.**

The pair $\left(\left[\oplus_{i=2}^{\infty}\ell_{i}^{2}\right]_{\ell_{2}},\ell_{\infty}\right)$ does not satisfy the Lp,o.

Proof.

We denote by $E_{i}$ and $\tilde{E}_{i}$ the natural embeddings from $\ell_{i}^{2}$ to $X$ and $(\ell_{i}^{2})^{*}$ to $X^{*}$. Also we denote by $P_{i}$ the natural projections from $\ell_{\infty}$ to the $i$th coordinate. For $f_{i}^{*}=(1,0)\in S_{(\ell_{i}^{2})^{*}}$, we define $T\in S_{\mathcal{L}(X,\ell_{\infty})}$ by $T(\cdot)=(\tilde{E}_{i}f_{i}^{*}(\cdot))_{i}$. Note that for each $z_{i}=\left(\frac{1}{2^{1/i}},\frac{1}{2^{1/i}}\right)\in S_{\ell_{i}^{2}}$, the element $z_{i}^{*}=\left(\frac{1}{2^{1-\frac{1}{i}}},\frac{1}{2^{1-\frac{1}{i}}}\right)$ is the unique norm-one functional so that $z_{i}^{*}(z_{i})=1$. This shows that $\tilde{E}_{i}z_{i}^{*}$ is the unique element in $S_{X^{*}}$ so that $\tilde{E}_{i}z_{i}^{*}(E_{i}z_{i})=1$, and then if an operator $S\in S_{\mathcal{L}(X,\ell_{\infty})}$ attains its norm at $E_{i}z_{i}$, then there exists $j_{0}\in\mathbb{N}$ and a modulus 1 scalar $c$ so that $P_{j_{0}}S=c\tilde{E}_{i}z_{i}^{*}$. From the construction, we see that

[TABLE]

for any modulus 1 scalar $c$ and $j\in\mathbb{N}$. This proves that $\left(\left[\oplus_{i=2}^{\infty}\ell_{i}^{2}\right]_{\ell_{2}},\ell_{\infty}\right)$ cannot satisfy the Lp,o ∎

Next we give some results on stability concerning properties Lp,o and Lo,p. Recall that a subspace $Z$ of a Banach space $X$ is one-complemented if $Z$ is the range of a norm-one projection on $X$.

Theorem 2.10**.**

Let $X,Y$ be Banach spaces, and let $Z$ be a one-complemented subspace $Z$ of $X$.

• (i)

If the pair $(X,Y)$ has the Lp,o, then so does $(Z,Y)$.

• (ii)

If the pair $(X,Y)$ has the Lo,p, then so does $(Z,Y)$.

Proof.

We denote by $E$ and $P$ the canonical embedding and projection between $Z$ and $X$, respectively.

(i). Let $\varepsilon>0$ and $T\in S_{\mathcal{L}(Z,Y)}$ be given. Assume that $z\in S_{Z}$ satisfy $\|T(z)\|>1-\eta(\varepsilon,T\circ P)$, where $\eta(\cdot,\cdot)$ is the function for the pair $(X,Y)$ having the Lp,o. Since $\|(T\circ P)(E(z))\|=\|T(z)\|$ and $\|T\circ P\|=\|T\|$, there exists $S\in S_{\mathcal{L}(X,Y)}$ such that $\|S(E(z))\|=1$ and $\|S-T\circ P\|<\varepsilon$. Since $\|S\circ E-T\|\leqslant\|S-T\circ P\|$, we finish the proof.

(ii). Let $\varepsilon>0$ and $z\in S_{Z}$ be given. Assume that $T\in S_{\mathcal{L}(Z,Y)}$ satisfy $\|T(z)\|>1-\eta(\varepsilon,E(z))$, where $\eta(\cdot,\cdot)$ is the function for the pair $(X,Y)$ having the Lo,p. Since $\|(T\circ P)(E(z))\|=\|T(z)\|$ and $\|T\circ P\|=\|T\|$, there exists $x\in S_{X}$ such that $\|x-E(z)\|<\varepsilon$ and $\|T\circ P(x)\|=1$. Since $\|P(x)-z\|\leqslant\|x-E(z)\|$, we finish the proof. ∎

Proposition 2.11**.**

Let $X$ and $Y$ be Banach spaces.

• (i)

If the pair $(X,Y)$ has the Lo,p for some Banach space $Y$, then so does $(X,\mathbb{K})$.

• (ii)

If the pair $(X,Y)$ has the Lp,o for some Banach space $Y$, then so does $(X,\mathbb{K})$.

Proof.

(i). Let $\varepsilon>0$ and $x\in S_{X}$ be given. By hypothesis, there is $\eta(\varepsilon,x)>0$ for the pair $(X,Y)$. Let $x^{*}\in X^{*}$ with $\|x^{*}\|=1$ be such that $|x^{*}(x)|>1-\eta(\varepsilon,x)$. Define $T\in\mathcal{L}(X,Y)$ by $T(z):=x^{*}(z)y_{0}$ for $z\in X$ and for a fixed $y_{0}\in S_{Y}$. Then, $\|T\|=\|x^{*}\|=1$ and $\|T(x)\|=|x^{*}(x)|>1-\eta(\varepsilon,x)$. So, there is $x_{0}\in S_{X}$ such that $\|T(x_{0})\|=|x^{*}(x_{0})|=1$ and $\|x_{0}-x\|<\varepsilon$. This proves that $(X,\mathbb{K})$ has the Lo,p.

(ii). Let $\varepsilon>0$ and $x^{*}\in X^{*}$ with $\|x^{*}\|=1$ be given. Again, define $T(z):=x^{*}(z)y_{0}$ for $z\in X$ and for a fixed $y_{0}\in S_{Y}$. Set $\eta(\varepsilon,x^{*}):=\eta(\varepsilon,T)>0$. Let $x_{0}\in S_{X}$ be such that $|x^{*}(x_{0})|>1-\eta(\varepsilon,x^{*})$. Then, $\|T(x_{0})\|>1-\eta(\varepsilon,T)$. So, there is $S\in\mathcal{L}(X,Y)$ with $\|S\|=1$ such that $\|S(x_{0})\|=1$ and $\|S-T\|<\varepsilon$. Let $y_{0}^{*}\in S_{Y^{*}}$ be such that $y_{0}^{*}(S(x_{0}))=\|S(x_{0})\|=1$. Set $x_{1}^{*}:=S^{*}y_{0}^{*}\in S_{X^{*}}$. Then, $|x_{1}^{*}(x_{0})|=1$ and $\|x_{1}^{*}-x^{*}\|<\varepsilon$. This means that the pair $(X,\mathbb{K})$ has the Lp,o. ∎

By Proposition 2.5, we know that the pairs $(\ell_{1}^{2},\mathbb{K})$ and $(\ell_{\infty}^{2},\mathbb{K})$ fails both Lo,p and Lp,o. So, as a consequence of Theorem 2.10 and Proposition 2.11, if $X$ has $\ell_{1}^{2}$ or $\ell_{\infty}^{2}$ as a one-complemented subspace, then the pair $(X,Y)$ cannot have neither Lo,p nor Lp,o for all Banach spaces $Y$. Hence, we have the following consequence.

Corollary 2.12**.**

Let $Y$ be a Banach space. The pairs $(\ell_{1},Y)$ and $(c_{0},Y)$ fail both Lo,p and Lp,o.

We finish the paper by discussing some of the relations between the Bishop-Phelps-Bollobás properties we mentioned so far. There are two more of them we would like to consider that we did not discuss in the present article. They are the local versions of the BPBp, which we denote by L△**, where $\triangle$ means that the $\eta$ depends on a fixed point $x$ or on a fixed operator $T$. A pair of Banach spaces $(X,Y)$ has the Lp if given $\varepsilon>0$ and $x\in S_{X}$, then there is $\eta(\varepsilon,x)>0$ such that whenever $T\in\mathcal{L}(X,Y)$ with $\|T\|=1$ satisfies $\|T(x)\|>1-\eta(\varepsilon,x)$, there are $S\in\mathcal{L}(X,Y)$ with $\|S\|=1$ and $x_{0}\in S_{X}$ such that

[TABLE]

On the other hand, $(X,Y)$ has the Lo if given $\varepsilon>0$ and $T\in S_{\mathcal{L}(X,Y)}$, there is $\eta(\varepsilon,T)>0$ such that whenever $x\in S_{X}$ satisfies $\|T(x)\|>1-\eta(\varepsilon,T)$, there are $S\in\mathcal{L}(X,Y)$ with $\|S\|=1$ and $x_{0}\in S_{X}$ such that (3) holds. For more information about these properties, we refer the reader to [16, Section 3]. In the next remark we compare the properties we have considered.

Remark 2.13**.**

We have the following observations.

1. (i)

All the implications below between the Bishop-Phelps-Bollobás properties hold.

BPBpp

[TABLE]

[TABLE]

[TABLE] Lp

[TABLE]

[TABLE]

[TABLE]

[TABLE] {}^{{}^{(1)}}$${}^{{}^{(2)}}$${}^{{}^{(4)}}$${}^{{}^{(5)}}$${}^{{}^{(8)}}$${}^{{}^{(9)}}$${}^{{}^{(6)}}$${}^{{}^{(7)}}$${}^{{}^{(3)}}$${}^{{}^{(10)}}(11)(12)

On the other hand, the reverse implications are not true. 2. (ii)

The BPBp does not imply L□,△**, where $\square$ and $\triangle$ can be $p$ or $o$. 3. (iii)

There is no relation between properties Lo,p and Lp,o. 4. (iv)

The Lp,p does not imply the Lp,o, but we do not know whether the Lp,o implies (or not) the Lp,p. 5. (v)

The Lo,o does not imply the Lo,p, but we do not know whether the Lo,p implies (or not) the Lo,o.

We briefly discuss the statements in the above remark. It is clear that all the implications in (i) are satisfied, so let us show that the reverse implications do not hold. In [16, Section 5] it is proved that the reverse implications of (2), (3), (5), (6), (8), (10), (11) and (12) do not hold. The reverse implication of (4) (respectively (9)) fails since, for instance, the pairs $(\ell_{1},\mathbb{K})$ or $(c_{0},\mathbb{K})$ have the Lo (respectively Lp) but fail the Lp,o (respectively Lo,p). To show that the reverse implication of (7) fails, just take a pair $(X,\mathbb{K})$ with $X$ reflexive and LUR but not uniformly convex. Analogously (reasoning with $X^{*}$ instead of $X$) we see that the reverse implication of (1) does not hold. To see (ii), just note that $(X,\mathbb{K})$ has the BPBp for every Banach space $X$, which is clearly not true for any of the properties L□,△**. For (iii), take $X$ a uniformly smooth Banach space with $\dim(X)\geqslant 2$. Then, we have that $(X,\ell_{\infty}^{2})$ has the BPBpp (see [13, Proposition 2.4]) and, consequently, the Lp,o, but fails the Lo,p in virtue of Proposition 2.7. This shows that the Lp,o does not imply the Lo,p. For the converse, take any finite-dimensional space $X$ which is strictly convex but not smooth. Then, $X^{*}$ cannot be strictly convex and by Corollary 2.2.(ii), the pair $(X,\mathbb{K})$ fails property Lp,o. On the other hand, by using Proposition 2.1.(i), it satisfies property Lo,p. So, the Lo,p cannot imply the Lp,o. Finally, for (iv) and (v), notice that the Lp,p cannot imply the Lp,o since $(\ell_{1}^{2},\mathbb{K})$ has the Lp,p (see [16, Proposition 2.9]) but fails the Lp,o (see Proposition 2.5). The same example shows that Lo,o does not imply the Lo,p.