Bishop-Phelps-Bollob\'as property for positive operators when the domain is $L_\infty $

TL;DR

This paper establishes that positive operators from $L_ abla$ to certain Banach lattices possess the Bishop-Phelps-Bollobás property, extending to pairs like $(c_0, Y)$, with optimality results for strictly monotone lattices.

Contribution

It proves the Bishop-Phelps-Bollobás property for positive operators from $L_ abla$ and $c_0$ to uniformly monotone Banach lattices, including optimality conditions.

Findings

Positive operators from $L_ abla$ to $Y$ have the Bishop-Phelps-Bollobás property.

The property also holds for the pair $(c_0, Y)$.

Results are optimal for strictly monotone Banach lattices.

Abstract

We prove that the class of positive operators from $L_\infty (\mu)$ to $Y$ has the Bishop-Phelps-Bollob\'as property for any positive measure $\mu$, 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.

Equations

$$\Vert T(u_0)\Vert =1,\Vert u_0-x_0\Vert <\epsilon \text{and}\Vert T-S\Vert <\epsilon .$$

$$\Vert f_1+f_2\Vert \le 1\text{and}\frac{1}{1+\delta (\frac{\epsilon }{3})}\le \Vert f_1-f_2\Vert ,$$

$$\Vert h_1+h_2\Vert =1\text{and}\Vert h_i-f_i\Vert <\epsilon \text{for}i=1,2.$$

$$C_1=\{t\in \Omega :f_2(t)\le f_1(t)\}\text{and}{C}_2=\{t\in \Omega :f_1(t)<f_2(t)\}.$$

$$|f_1-f_2|+h=f_1+f_2.$$

$$\Vert f_1+f_2\Vert$$

$$\displaystyle=\Bigl{(}1+\delta\Bigl{(}\frac{\varepsilon}{3}\Bigr{)}\Bigr{)}\|\;|f_{1}-f_{2}|\;\|.$$

$$\|h\|\leq\frac{\varepsilon}{3}\bigl{\|}|f_{1}-f_{2}|\bigr{\|}=\frac{\varepsilon}{3}\|f_{1}-f_{2}\|\leq\frac{\varepsilon}{3}\|f_{1}+f_{2}\|\leq\frac{\varepsilon}{3}.$$

$$\Vert f_1{\chi }_{C_2}\Vert \le \frac{\epsilon }{6}\text{and}\Vert f_2{\chi }_{C_1}\Vert \le \frac{\epsilon }{6}.$$

$$\Vert g_1+g_2\Vert \le 1.$$

$$\Vert g_1-f_1\Vert =\Vert f_1{\chi }_{C_1}-f_1\Vert =\Vert f_1{\chi }_{C_2}\Vert \le \frac{\epsilon }{6}.$$

$$\Vert g_2-f_2\Vert \le \frac{\epsilon }{6}.$$

$$\Vert g_1+g_2\Vert$$

$$\ge \Vert f_1-f_2\Vert -\frac{\epsilon }{3}\text{(by eqrefg1-f1-small and eqrefg2-f2-small)}$$

$$\ge \frac{1}{1+\delta (\frac{\epsilon }{3})}-\frac{\epsilon }{3}$$

$$\ge \frac{1}{1+\frac{\epsilon }{3}}-\frac{\epsilon }{3}>0.$$

$$\Vert h_i-g_i\Vert$$

$$\displaystyle=\frac{\|g_{i}\|}{\|g_{1}+g_{2}\|}\bigl{|}1-\bigl{\|}g_{1}+g_{2}\bigr{\|}\;\bigr{|}$$

$$\displaystyle\leq 1-\bigl{\|}g_{1}+g_{2}\bigr{\|}{\ \ \ \ }\text{(by \eqref{g1+g2-less-1})}$$

$$\le 1-\frac{1}{1+\delta (\frac{\epsilon }{3})}+\frac{\epsilon }{3}\text{(by eqrefg1-g2)}$$

$$<\frac{2\epsilon }{3}.$$

$$\Vert h_i-f_i\Vert \le \Vert h_i-g_i\Vert +\Vert g_i-f_i\Vert <\frac{2\epsilon }{3}+\frac{\epsilon }{6}<\epsilon .$$

$$\frac{1}{1+\delta (\frac{\epsilon }{18})}<\frac{1}{1+\delta ({\eta }^2)}-3\eta .$$

$$\Vert S(f_0)\Vert >\frac{1}{1+\delta ({\eta }^2)}.$$

$$A=\{t\in {\Omega }_1:-1\le f_0(t)<-1+\eta \},B=\{t\in {\Omega }_1:1-\eta <f_0(t)\le 1\}$$

$$C=\{t\in {\Omega }_1:|f_0(t)|\le 1-\eta \}.$$

$$\Vert S(|f_0|+\eta {\chi }_C)\Vert$$

$$\displaystyle<\|S(f_{0})\|\bigl{(}1+\delta(\eta^{2})\bigr{)}$$

$$\displaystyle\leq\|S(|f_{0}|)\|\bigl{(}1+\delta(\eta^{2})\bigr{)}.$$

$$\Vert S({\chi }_C)\Vert \le \eta .$$

$$\Vert S({\chi }_A)+S(f_0{\chi }_A)\Vert \le \eta \text{and }\Vert S(f_0{\chi }_B)-S({\chi }_B)\Vert \le \eta .$$

$$\Vert S({\chi }_B)-S({\chi }_A)\Vert$$

$$\ge \Vert S(f_0)\Vert -\Vert S(f_0{\chi }_C)\Vert -2\eta \text{(by eqrefS-f0A-close-SA)}$$

$$>\frac{1}{1+\delta ({\eta }^2)}-\Vert S({\chi }_C)\Vert -2\eta$$

Full text

The first author was supported by Junta de Andalucía grant FQM–185 and also by Spanish MINECO/FEDER grant PGC2018-093794-B-I00. The second author was supported by a grant from IPM

1. Introduction

In 1961 Bishop and Phelps proved that every continuous linear functional on a Banach space can be approximated by norm attaining functionals [10]. Since then a lot of attention has been devoted to extend Bishop-Phelps theorem in the setting of operators on Banach spaces (see [2]). On the other hand Bollobás proved a “quantitative version” of that result in 1970 [11]. Before stating this result we introduce some notation. By $B_{X}$, $S_{X}$ and $X^{*}$ we denote the closed unit ball, the unit sphere and the topological dual of a Banach space $X,$ respectively. If $X$ and $Y$ are both real or complex Banach spaces, $L(X,Y)$ denotes the space of (bounded linear) operators from $X$ to $Y,$ endowed with its usual operator norm.

Bishop-Phelps-Bollobás theorem (see [12, Theorem 16.1] or [13, Corollary 2.4]). Let $X$ be a Banach space and $0<\varepsilon<1$. Given $x\in B_{X}$ and $x^{*}\in S_{X^{*}}$ with $|1-x^{*}(x)|<\frac{\varepsilon^{2}}{2}$, there are elements $y\in S_{X}$ and $y^{*}\in S_{X^{*}}$ such that $y^{*}(y)=1$, $\|y-x\|<\varepsilon$ and $\|y^{*}-x^{*}\|<\varepsilon$.

In 2008, Acosta, Aron, García and Maestre defined the Bishop-Phelps-Bollobás property for operators between Banach spaces [4].

Definition 1.1** ([4, Definition 1.1]).**

Let $X$ and $Y$ be either real or complex Banach spaces. The pair $(X,Y)$ is said to have the Bishop-Phelps-Bollobás property for operators if for every $0<\varepsilon<1$ there exists $0<\eta(\varepsilon)<\varepsilon$ such that for every $S\in S_{L(X,Y)}$, if $x_{0}\in S_{X}$ satisfies $\|S(x_{0})\|>1-\eta(\varepsilon)$, then there exist an element $u_{0}\in S_{X}$ and an operator $T\in S_{L(X,Y)}$ satisfying the following conditions

[TABLE]

Since then a number of interesting results related to this property have been obtained (see [3]).

Very recently in [8], the authors introduced the notion of Bishop-Phelps-Bollobás property for positive operators between two Banach lattices. Let us mention that the only difference between this property and the previous one is that in the new property the operators appearing in Definition 1.1 are positive. In the same paper it is proved that the pairs $(c_{0},L_{1}(\nu))$ and $(L_{\infty}(\mu),L_{1}(\nu))$ have the Bishop-Phelps-Bollobás property for positive operators for any positive measures $\mu$ and $\nu$ (see [8, Theorems 3.1 and 2.5]).

In this paper we show a far reaching extension of those results. More precisely we prove that the pair $(c_{0},Y)$ has the Bishop-Phelps-Bollobás property for positive operators whenever $Y$ is a uniformly monotone Banach lattice (see Corollary 3.3). We also show that the pair $(L_{\infty}(\mu),Y)$ has the Bishop-Phelps-Bollobás property for positive operators for any positive measure $\mu$ if $Y$ is a uniformly monotone Banach lattice with a weak unit (see Corollary 2.6). Notice that the last assumption is very mild. For instance, separable Banach lattices have a weak unit (see [9, Lemma 3, p. 367]).

We also remark that not every Banach function space $Y$ satisfies that the pair $(c_{0},Y)$ has the Bishop-Phelps-Bollobás property for positive operators (see [8, Example 3.2]). It is worth also to mention that it is not known whether or not the pair $(c_{0},\ell_{1})$ has the Bishop-Phelps-Bollobás property for operators in the real case. For some partial results when the domain is $c_{0}$ see [15],[7], [6] and [5]. The paper [16] contains a positive result for the pair $(L_{\infty}(\mu),Y)$, whenever $Y$ is a uniformly convex Banach space.

In the last section of the paper we show under very mild assumptions that anytime that a pair of Banach lattices $(X,Y)$ has the BPBp for positive operators, then $Y$ is indeed uniformly monotone, whenever $X$ admits non trivial $M$-summands and $Y$ is strictly monotone (see Proposition 4.3). As a consequence, the geometrical assumption on the range space $Y$ in the main results of sections 2 and 3 are optimal in case that $Y$ is strictly monotone (see Corollary 4.4).

We remark that throughout this paper we consider only real Banach spaces.

2. Bishop-Phelps-Bollobás property for positive operators from $L_{\infty}$ to a uniformly monotone Banach lattice

We begin by recalling some definitions and the appropriate notion of Bishop-Phelps-Bollobás property for positive operators. For the terminology and basic facts related to Banach lattices see, for instance, [1], [9] and [17].

An ordered vector space is a real vector space $X$ equipped with an order relation $\leq$ that is compatible with the algebraic structure of $X$. An ordered vector space is a Riesz space if every pair of vectors has a least upper bound and a greatest lower bound. A norm $\|\ \|$ on a Riesz space $X$ is said to be * a lattice norm * whenever $|x|\leq|y|$ implies $\|x\|\leq\|y\|$. A normed Riesz space is a Riesz space equipped with a lattice norm. A Banach lattice is a normed Riesz space whose norm is complete. A positive element $e$ in a Banach lattice $X$ is a weak unit if $x\wedge e=0$ for some $x\in X$ implies that $x=0.$

A Banach lattice $E$ is uniformly monotone if for every $\varepsilon>0$ there is $\delta(\varepsilon)>0$ such that whenever $x\in S_{E}$, $y\in E$ and $x,y\geq 0$ the condition $\|x+y\|\leq 1+\delta(\varepsilon)$ implies that $\|y\|\leq\varepsilon$. A Banach lattice $X$ is said to be order continuous whenever $(x_{\alpha})\downarrow 0$ in $X$ implies $(\|x_{\alpha}\|)\to 0$.

In case that $(\Omega,\mu)$ is a measure space, we denote by $L^{0}(\mu)$ the space of (equivalence classes of $\mu$-a.e. equal) real valued measurable functions on $\Omega$. We say that a Banach space $X$ is a Banach function space on $(\Omega,\mu)$ if $X$ is an ideal in $L^{0}(\mu)$ and whenever $x,y\in X$ and $|x|\leq|y|$ a.e., then $\|x\|\leq\|y\|.$

An operator $T:X\to Y$ between two ordered vector spaces is called positive if $x\geq 0$ implies $Tx\geq 0$.

Let us notice that in case that $Y$ is a uniformly monotone Banach lattice it follows from the definition that the function $\delta$ satisfies $\delta(t)\leq t$ for every positive real $t$.

Definition 2.1** ([8, Definition 2.2]).**

Let $X$ and $Y$ be Banach lattices. The pair $(X,Y)$ is said to have the Bishop-Phelps-Bollobás property for positive operators if for every $0<\varepsilon<1$ there exists $0<\eta(\varepsilon)<\varepsilon$ such that for every $S\in S_{L(X,Y)}$, such that $S\geq 0$, if $x_{0}\in S_{X}$ satisfies $\|S(x_{0})\|>1-\eta(\varepsilon)$, then there exist an element $u_{0}\in S_{X}$ and a positive operator $T\in S_{L(X,Y)}$ satisfying the following conditions

[TABLE]

Remark 2.2**.**

In case that the pair $(X,Y)$ satisfies the previous definition, if the element $x_{0}$ is positive, then the element $u_{0}$ can also be chosen positive.

Proof.

Assume that $(X,Y)$ has the BPBp for positive operators and assume that $S\in S_{L(X,Y)}$ and $x_{0}\in S_{X}$ satisfy the assumptions in Definition 2.1 and $x_{0}$ is also positive. So there exists a pair $(T,u_{0})\in S_{L(X,Y)}\times S_{X}$ such that $T$ is positive and satisfying

[TABLE]

We will check that the positive element $|u_{0}|$ also satisfies the desired conditions.

Notice that from triangle inequality we have $\||u_{0}|-|x_{0}|\|\leq\|u_{0}-x_{0}\|<\varepsilon$, so since $x_{0}$ is positive we conclude $\||u_{0}|-x_{0}\|\leq\|u_{0}-x_{0}\|<\varepsilon$. On the other hand since the operator $T$ is positive, $|T(u_{0})|\leq T(|u_{0}|).$ Hence $1=\|T(u_{0})\|\leq\|T(|u_{0}|)\|\leq\|T\|=1$, so $\|T(|u_{0}|)\|=1.$ Therefore the element $|u_{0}|\in S_{X}$ satisfies the desired conditions. ∎

Next we show some technical results that will be useful later. Throughout the rest of the section, if $(\Omega,\mu)$ is a measure space, we denote by ${\mathds{1}}$ the constant function equal to $1$ on $\Omega$. Since an element $f$ in $B_{L_{\infty}(\mu)}$ satisfies that $|f|\leq{\mathds{1}}$ a.e., it is clear that a positive operator from $L_{\infty}(\mu)$ to any other Banach lattice satisfies the next assertion.

Lemma 2.3**.**

Let $\mu$ be a positive measure and $T$ a positive operator from $L_{\infty}(\mu)$ to some Banach lattice $Y.$ Then $\|T\|=\|T({\mathds{1}})\|$.

The next result extends [8, Lemma 2.4], where the analogous result was proved for $L_{1}(\mu).$

Lemma 2.4**.**

Let $Y$ be a uniformly monotone Banach function space and $0<\varepsilon<1$. Assume that $f_{1}$ and $f_{2}$ are positive elements in $Y$ such that

[TABLE]

where $\delta$ is the function satisfying the definition of uniform monotonicity for $Y.$ Then there are two positive functions $h_{1}$ and $h_{2}$ in $Y$ with disjoint supports satisfying that

[TABLE]

Proof.

Assume that $Y$ is a Banach function space on the measure space $(\Omega,\mu).$ We consider the partition of $\Omega$ given by the following sets

[TABLE]

Clearly $C_{1}$ and $C_{2}$ are measurable sets. The function $h$ given by h=2\bigl{(}f_{1}\chi_{C_{2}}+f_{2}\chi_{C_{1}}) belongs to $Y$ since $Y$ is a Banach function space on $(\Omega,\mu)$. It is also clear that $h$ is positive and it is satisfied that

[TABLE]

Clearly we have that

[TABLE]

So from (2.1) it follows that

[TABLE]

As a consequence

[TABLE]

Now define $g_{i}=f_{i}\chi_{C_{i}}$ for $i=1,2$. Notice that $g_{1}$ and $g_{2}$ are positive functions with disjoint supports, belong to $Y$ and also satisfy

[TABLE]

By (2.2) the function $g_{1}$ satisfies

[TABLE]

By using the same argument we also obtain that

[TABLE]

It is also satisfied that

[TABLE]

For $i=1,2$ we can define the function $h_{i}$ by $h_{i}=\frac{g_{i}}{\|g_{1}+g_{2}\|}.$ Clearly $h_{1}$ and $h_{2}$ are positive functions in $Y$ with disjoint supports satisfying also that $\|h_{1}+h_{2}\|=1$.

For $i=1,2$ we also have that

[TABLE]

By using (2), (2.4) and (2.5) we can estimate the distance between $h_{i}$ and $f_{i}$ for $i=1,2$ as follows

[TABLE]

This finishes the proof. ∎

Theorem 2.5**.**

The pair $(L_{\infty}(\mu),Y)$ has the Bishop-Phelps–Bollobás property for positive operators, for any positive measure $\mu$, whenever $Y$ is a uniformly monotone Banach function space. The function $\eta$ satisfying Definition 2.1 depends only on the modulus of uniform monotonicity of $Y$.

Proof.

Assume that $(\Omega_{1},\mu)$ is a measure space and $Y$ is a Banach function space on $(\Omega_{2},\nu).$ Let $0<\varepsilon<1$ and $\delta$ be the function satisfying the definition of uniform monotonicity for the Banach function space $Y.$ Choose a real number $\eta$ such that $0<\eta=\eta(\varepsilon)<\frac{\varepsilon}{18}$ and satisfying also

[TABLE]

Assume that $f_{0}\in S_{L_{\infty}(\mu)},S\in S_{L(L_{\infty}(\mu),Y)}$ and $S$ is a positive operator such that

[TABLE]

We can assume without loss of generality that $|f_{0}|\leq 1.$ We define the sets $A,B$ and $C$ given by

[TABLE]

and

[TABLE]

Clearly $\{A,B,C\}$ is a partition of $\Omega_{1}$ into measurable sets. We clearly have that $|f_{0}|+\eta\chi_{C}\in S_{L_{\infty}(\mu)}$. By using that $S$ is a positive operator we have that

[TABLE]

In view of the uniform monotonicity of $Y$ the previous inequality implies that $\|S(\eta\chi_{C})\|\leq\eta^{2}$ and so

[TABLE]

From the definition of the sets $A$ and $B$ we have that $\|\chi_{A}+f_{0}\chi_{A}\|_{\infty}\leq\eta$ and $\|f_{0}\chi_{B}-\chi_{B}\|_{\infty}\leq\eta$ and so

[TABLE]

We clearly obtain that

[TABLE]

Since $S$ is a positive operator and $\|S(\chi_{A})+S(\chi_{B})\|\leq 1$, in view of (2) we can apply Lemma 2.4. Hence there are two positive functions $h_{1}$ and $h_{2}$ in $Y$ satisfying the following conditions

[TABLE]

[TABLE]

Hence

[TABLE]

and

[TABLE]

Now we define the operator $U:L_{\infty}(\mu)\longrightarrow Y$ as follows

[TABLE]

Since $Y$ is a Banach function space and $S\in L(L_{\infty}(\mu),Y),$ $U$ is well defined and belongs to $L(L_{\infty}(\mu),Y).$ The operator $U$ is positive since $S$ is positive. It also satisfies that

[TABLE]

Hence

[TABLE]

so $U\neq 0$.

Finally we define $T=\frac{U}{\|U\|}$. Since $U$ is a positive operator, $T$ is also positive. Of course $T\in S_{L(L_{\infty}(\mu),Y)}$ and also satisfies

[TABLE]

The function $f_{1}$ given by $f_{1}=\chi_{B}-\chi_{A}+f_{0}\chi_{C}$ belongs to $S_{L_{\infty}(\mu)}$ and satisfies that

[TABLE]

We clearly have that

[TABLE]

Since $U$ is a positive operator it satisfies that

[TABLE]

For each $t\in\Omega_{2}$ we obtain that

[TABLE]

Since $Y$ is a Banach function space we conclude that

[TABLE]

By (2.19) and (2), since $T$ attains its norm at $f_{1}$, the proof is finished ∎

The previous result was proved in [8, Theorem 2.5] in case that the range is a $L_{1}$ space, so Theorem 2.5 is already a far reaching extension of that result.

Our purpose now is to obtain a version of Theorem 2.5 for some abstract Banach lattices. In order to get this result, we notice that every uniformly monotone Banach lattice is order continuous (see [9, Theorem 21, p. 371] and [17, Proposition 1.a.8]). It is also known that any order continuous Banach lattice with a weak unit is order isometric to a Banach function space (see [17, Theorem 1.b.14]). From Theorem 2.5 and the previous argument we deduce the following result.

Corollary 2.6**.**

The pair $(L_{\infty}(\mu),Y)$ has the Bishop-Phelps-Bollobás property for positive operators, for any positive measure $\mu$, whenever $Y$ is a uniformly monotone Banach lattice with a weak unit. Moreover, the function $\eta$ satisfying Definition 2.1 depends only on the modulus of uniform monotonicity of $Y$.

3. Bishop-Phelps–Bollobás property for positive operators in case that the domain is $c_{0}$.

In this section we show parallel results to Theorem 2.5 and Corollary 2.6 in case that the domain space is $c_{0}$. We begin with a technical result whose proof is straightforward.

Lemma 3.1**.**

Let $Y$ be a Banach lattice and $T\in L(c_{0},Y)$ be a positive operator. Then the following assertions are satisfied:

$\|T\|=\sup\{\|T(\sum_{k=1}^{n}e_{i})\|:n\in{\mathbb{N}}\}=\sup\{\|T(\sum_{k=1}^{n}e_{i})\|:n\geq N\}$, for all positive integer $N.$ 2. 2)

$\sup\{\|T(x\chi_{C})\|:x\in B_{c_{0}}\}=\sup\{\|T(\chi_{C\cap\{k\in{\mathbb{N}}:k\leq n\}})\|:n\in{\mathbb{N}},n\geq N\}$* for all $C\subset{\mathbb{N}}$ and all positive integer $N.$*

Theorem 3.2**.**

The pair $(c_{0},Y)$ satisfies the Bishop-Phelps–Bollobás property for positive operators, for any uniformly monotone Banach function space $Y.$ Moreover, the function $\eta$ satisfying Definition 2.1 depends only on the modulus of uniform monotonicity of $Y$.

Proof.

The proof of this result is similar to the proof of Theorem 2.5. We include the details of the proof for the sake of completeness.

Assume that $Y$ is a Banach function space on the measure space $(\Omega,\mu).$

Let $0<\varepsilon<1,$ and assume that $Y$ satisfies the definition of uniform monotonicity with the function $\delta.$ Choose a positive real number such that $\eta<\frac{\varepsilon}{18}$ satisfying also

[TABLE]

Assume that $x_{0}\in S_{c_{0}},S\in S_{L(c_{0},Y)}$ and $S$ is a positive operator such that

[TABLE]

Consider the sets $A,B$ and $C$ defined by

[TABLE]

and

[TABLE]

Clearly $\{A,B,C\}$ is a partition of ${\mathbb{N}}$. Also the subsets $A$ and $B$ are finite sets, so $\chi_{A}$ and $\chi_{B}$ belong to $c_{0}$. For a positive integer $n$ denote by $C_{n}=C\cap\{k\in{\mathbb{N}}:k\leq n\}$. We clearly have that $|x_{0}|+\eta\chi_{C_{n}}\in S_{c_{0}}$ for all $n\in{\mathbb{N}}.$ Since $S$ is a positive operator for each positive integer $n$ it is satisfied that

[TABLE]

In view of the uniform monotonicity of $Y$ the previous inequality implies that $\|S(\eta\chi_{C_{n}})\|\leq\eta^{2}$ for all $n\in{\mathbb{N}}$ and so

[TABLE]

From Lemma 3.1 we deduce that

[TABLE]

From the definition of the sets $A$ and $B$ we also have that $\|\chi_{A}+x_{0}\chi_{A}\|_{\infty}\leq\eta$ and $\|x_{0}\chi_{B}-\chi_{B}\|_{\infty}\leq\eta,$ so

[TABLE]

We clearly obtain that

[TABLE]

Since $S$ is a positive operator and $\|S(\chi_{A})+S(\chi_{B})\|\leq 1$, in view of (3) we can apply Lemma 2.4. Hence there are two positive functions $g_{1}$ and $g_{2}$ in $Y$ satisfying the following conditions

[TABLE]

[TABLE]

Hence

[TABLE]

and

[TABLE]

Now we define the operator $R:c_{0}\longrightarrow Y$ as follows

[TABLE]

Since $Y$ is a Banach function space and $S\in L(c_{0},Y)$, $R$ is well defined and belongs to $L(c_{0},Y)$. The operator $R$ is positive since $S$ is positive. By using Lemma 3.1, for any element $x\in B_{c_{0}}$ we have that

[TABLE]

As a consequence

[TABLE]

so $R\neq 0$.

We put $T=\frac{R}{\|R\|}$. The operator $T$ is positive since $R$ is positive. Of course $T\in S_{L(c_{0},Y)}$. Now we estimate the distance from $T$ to $S$ as follows

[TABLE]

The element $u_{0}$ given by $u_{0}=\chi_{B}-\chi_{A}+x_{0}\chi_{C}$ belongs to the unit sphere of $c_{0}$ since $x_{0}\in S_{c_{0}}.$ It also satisfies

[TABLE]

Since $g_{1}$ and $g_{2}$ have disjoint supports, $Y$ is a Banach function space and $R$ is a positive operator, we also have that

[TABLE]

Hence the operator $T$ also attains its norm at $u_{0}$. In view of (3) and (3.11) the proof is finished.

∎

Corollary 3.3**.**

The pair $(c_{0},Y)$ has the Bishop-Phelps–Bollobás property for positive operators, for any uniformly monotone Banach lattice $Y.$ Moreover, the function $\eta$ satisfying Definition 2.1 depends only on the modulus of uniform monotonicity of $Y$.

Proof.

By using the same argument of Corollary 2.6, in view of Theorem 3.2, we obtain the statement when $Y$ is a uniformly monotone Banach lattice with a weak unit. Notice also that in Theorem 3.2 the function $\eta$ appearing in Definition 2.1 depends only on the modulus of uniform monotonicity of the Banach function space on the range.

Assume now that $Y$ is a uniformly monotone Banach lattice with modulus of uniform monotonicity $\delta$. Since $c_{0}$ is separable, if $T\in L(c_{0},Y)$, the space $T(c_{0})$ is also separable. It is well known that the Banach lattice $X$ generated by $T(c_{0})$ is a separable Banach lattice of $Y$ and so, it also satisfies the definition of uniform monotonicity with the function $\delta$. By [9, Lemma 3, p. 367] $X$ has a weak unit. By the previous arguments, the pair $(c_{0},X)$ satisfies the Bishop-Phelps–Bollobás property for positive operators with a function $\eta$ depending only on $\delta$. As a consequence, the pair $(c_{0},Y)$ also has the Bishop-Phelps–Bollobás property for positive operators. ∎

Kim proved that the pair $(c_{0},Y)$ has the Bishop-Phelps-Bollobás property for operators when $Y$ is uniformly convex (see [15, Corollary 2.6]). Notice that the previous result is a version of that result for positive operators.

4. Results are optimal when the range is strictly monotone

Our intention now is to show that under some mild assumption on the range space, Corollaries 2.6 and 3.3 are optimal. For this end we need some preliminary results.

Lemma 4.1**.**

Let $M,N$ and $Y$ be normed spaces. Assume that $T\in{L(M\oplus_{\infty}N,Y)}$ and $m+n\in S_{M\oplus_{\infty}N}$ satisfy that $\|n\|<1$ and $\|T(m+n)\|=\|T\|.$ Then $\|T(m)\|=\|T\|$.

Proof.

If we put $t=\|n\|$, then $0\leq t<1$. In case that $t=0$ it is clear that $T$ attains its norm at $m$. Otherwise $0<t<1$, the element $m+\frac{1}{t}n$ belongs to the unit sphere of $M\oplus_{\infty}N$ and

[TABLE]

Since $\|T(m+n)\|=\|T\|,$ and the function $x\mapsto\|T(x)\|$ is convex on $M\oplus N$, we conclude that $\|T(m)\|=\|T\|$. ∎

Next result is probably known, but we did not find a reference in the literature and we include it for the sake of completeness.

Proposition 4.2**.**

Let $Y$ be a Banach lattice. The following conditions are equivalent.

$Y$* is uniformly monotone.* 2. 2)

For every $0<\varepsilon<1$, there is $\eta(\varepsilon)>0$ satisfying

[TABLE] 3. 3)

For every $0<\varepsilon<1$, there is $\eta(\varepsilon)>0$ satisfying

[TABLE]

Moreover, if 2) is satisfied, $Y$ is uniformly monotone with $\delta(\varepsilon)=\eta(\frac{\varepsilon}{2})$. In case that $Y$ is uniformly monotone with $\delta(\varepsilon)$ conditions 2) and 3) are satisfied with $\eta(\varepsilon)=\frac{\delta(\varepsilon)}{1+\delta(\varepsilon)}.$

Proof.

1. $\Rightarrow$ 2)

Assume that $Y$ is uniformly monotone with $\delta(\varepsilon)$. Assume that $0<\varepsilon<1$ and $u$ and $v$ are elements in $Y$ such that

[TABLE]

Put $x=v-u,y=u$. So $x,y\geq 0$ and

[TABLE]

Hence from uniform monotonicity of $Y$ we conclude $\|u\|=\|y\|\leq\varepsilon\|x\|\leq\varepsilon.$

2. $\Rightarrow$ 3)

Trivial

3. $\Rightarrow$ 1)

Assume that $0<\varepsilon<1$, $x\in S_{Y},$ $y\in Y$ satisfy that $x,y\geq 0$ and $\|x+y\|\leq 1+\delta(\varepsilon)=1+\eta(\frac{\varepsilon}{2})$. Put $u=y,v=x+y$. So $0\leq u\leq v$ and

[TABLE]

Hence from the assumptions we conclude $\|y\|=\|u\|\leq\frac{\varepsilon}{2}\|v\|\leq\frac{\varepsilon}{2}(1+\eta(\frac{\varepsilon}{2}))\leq\varepsilon.$ So $Y$ is uniformly monotone with $\delta(\varepsilon)=\eta(\frac{\varepsilon}{2})$. ∎

Proposition 4.3**.**

Let $X$ be a Banach lattice, $M$ and $N$ be non zero Banach sublattices of $X$ such that $X=M\oplus_{\infty}N$ and the canonical projections from $X$ to $M$ and $N$ are positive operators. If $Y$ is a strictly monotone Banach lattice and the pair $(X,Y)$ has the Bishop-Phelps–Bollobás property for positive operators then $Y$ is uniformly monotone.

Proof.

We will show that $Y$ satisfies condition 2) in Proposition 4.2. Let $0<\varepsilon<1$. Let take elements $u$ and $v$ in $Y$ such that

[TABLE]

where $\eta$ is the function satisfying the definition of BPBp for positive operators for the pair $(X,Y)$.

Since $M$ and $N$ are non zero Banach sublattices, there are positive elements $m_{0}\in S_{M}$ and $n_{0}\in S_{N}$. By using Hahn-Banach theorem for Banach lattices and positive elements (see [18, Theorem 39.3]), there are positive functionals $m_{0}^{*}\in S_{M^{*}}$ and $n_{0}^{*}\in S_{N^{*}}$ such that $m_{0}^{*}(m_{0})=1=n_{0}^{*}(n_{0}).$

Let $P$ and $Q$ be the canonical projections from $X$ to $M$ and $N,$ respectively. Consider the operator $S$ from $X$ to $Y$ given by

[TABLE]

Since $v-u$ and $u$ are positive elements in $Y,$ the functionals $m_{0}^{*}$ and $n_{0}^{*}$ are positive on $M$ and $N,$ respectively, and the projections $P$ and $Q$ are positive operators on $X$, then $S$ is a positive operator from $X$ to $Y$.

By using the assumptions on $X$, the fact that the functionals $m_{0}^{*}$ and $n_{0}^{*}$ belong to the unit sphere of $M^{*}$ and $N^{*}$ respectively, since $v-u$ and $u$ are positive elements in $Y,$ for any element $x\in B_{X}$ we have that

[TABLE]

Since $m_{0}+n_{0}\in S_{X}$ and $S(m_{0}+n_{0})=v\in S_{Y}$, we deduce that $S\in S_{L(X,Y)}.$ Notice also that $\|S(m_{0})\|=\|v-u\|>1-\eta(\varepsilon)$. By using that the pair $(X,Y)$ has the BPBp for positive operators and Remark 2.2, there exist a positive operator $T\in S_{L(X,Y)}$ and a positive element $x_{1}\in S_{X}$ satisfying

[TABLE]

If we write $m_{1}=P(x_{1})$ and $n_{1}=Q(x_{1})$, since $\|n_{1}\|=\|Q(x_{1}-m_{0})\|\leq\|x_{1}-m_{0}\|<\varepsilon<1,$ from Lemma 4.1 we conclude that $\|T(m_{1})\|=1$.

Since $T$ and $P$ are positive operators and $x_{1}$ is a positive element in $X$, we have that

[TABLE]

By using that $Y$ is strictly monotone we obtain that $T(n_{0})=0.$ As a consequence,

[TABLE]

In view of Proposition 4.2 we proved that $Y$ is uniformly monotone. ∎

As a consequence of Proposition 4.3 and Corollaries 3.3 and 2.6 we deduce the following result, which is a version of the one obtained by Kim that asserts that a Banach space $Y$ is uniformly convex whenever it is strictly convex and the pair $(c_{0},Y)$ has the Bishop-Phelps–Bollobás property for operators ([15, Theorem 2.7]).

It is worth to point out that a Banach lattice is strictly monotone whenever it is strictly convex. Analogously uniform convexity implies uniform monotonicity (see [14, Theorem 1], for instance). Notice that the converse results do not hold since every $L_{1}(\mu)$ such that $\dim L_{1}(\mu)>1$ is uniformly monotone, but it is not strictly convex.

Corollary 4.4**.**

Let $Y$ be a strictly monotone Banach lattice. Then the pair $(c_{0},Y)$ has the BPBp for positive operators if and only if $Y$ is uniformly monotone. 2. 2)

Let $\mu$ be a positive measure such that $\dim L_{\infty}(\mu)>1$ and let $Y$ be a strictly monotone Banach lattice. If the pair $(L_{\infty}(\mu),Y)$ has the BPBp for positive operators, then $Y$ is uniformly monotone. In case that $Y$ has a weak unit the converse is also true.