Strong subdifferentiability and local Bishop–Phelps–Bollobás properties

Abstract

Some local versions of the Bishop–Phelps–Bollobás property for operators have been recently presented in Dantas et al. (J Math Anal Appl 468(1):304–323, 2018). In the present article, we continue studying these properties for multilinear mappings. We show some differences between the local and uniform versions of these and also provide some interesting examples which show that this study is not just a mere generalization of the linear case. As a consequence of our results, we get that, for $2<p,q<\infty$, the norm of the projective tensor product $\ell _p \hat{\otimes }_{\pi } \ell _{q}$ is strongly subdifferentiable. Moreover, we present necessary and sufficient conditions for the norm of a Banach space Y to be strongly subdifferentiable through the study of these properties for bilinear mappings on $\ell _1^N \times Y$.

1 Introduction

In Banach space theory, it is well-known that the set of all norm attaining continuous linear functionals defined on a Banach space X is dense in its topological dual space $X^*$. This is the famous Bishop–Phelps theorem [7]. In 1970, this result was strengthened by Bollobás, who proved a quantitative version in the following sense: if a norm-one linear functional $x^*$ almost attains its norm at some x, then, near to $x^*$ and x, there are, respectively, a new norm-one functional $y^*$ and a new point y such that $y^*$ attains its norm at y (see [8, Theorem 1]). Nowadays, this result is known as the Bishop–Phelps–Bollobás theorem and it has been used as an important tool in the study of Banach spaces and operators. For example, it was used to prove that the numerical radius of a continuous linear operator is the same as its adjoint in [8].

It is natural to ask whether the Bishop–Phelps and Bishop–Phelps–Bollobás theorems hold also for bounded linear operators. In 1963, Lindenstrauss gave the first example of a Banach space X such that the set of all norm attaining operators on X is not dense in the set of all bounded linear operators (see [33, Proposition 5]). On the other hand, he studied some conditions on the involved Banach spaces in order to get a Bishop–Phelps type theorem for operators. For instance, he proved that the set of all operators whose second adjoint attain their norms is dense, so, in particular, if X is a reflexive Banach space, then the set of all norm attaining operators is dense for arbitrary target spaces (actually, this result was extended by Bourgain in [10, Theorem 7] by showing that the Radon–Nikodým property implies the same result). This topic has been considered by many authors and we refer the reader to the survey paper [1] for more information and background about denseness of norm attaining operators. On the other hand, Acosta et al. [3] studied the vector-valued case of the Bishop–Phelps–Bollobás theorem and introduced the Bishop–Phelps–Bollobás property.

Now we introduce the notation and necessary preliminaries. Let N be a natural number. We use capital letters $X, X_1, \ldots , X_N, Y$ for Banach spaces over a scalar field ${\mathbb {K}}$ which can be the field of real numbers ${\mathbb {R}}$ or the field of complex numbers ${\mathbb {C}}$. The closed unit ball and the unit sphere of X are denoted by $B_X$ and $S_X$, respectively. The topological dual space of X is denoted by $X^*$ and ${\mathcal {L}} (X_1, \ldots , X_N; Y)$ stands for the set of all bounded N-linear mappings from $X_1 \times \cdots \times X_N$ into Y. For the convenience, if $X_1 = \cdots = X_N = X$, then we use the shortened notation ${\mathcal {L}} (^N X; Y)$. When $N = 1$, we have the set of all bounded linear operators from X into Y, which we denote simply by ${\mathcal {L}}(X; Y)$.

We say that an N-linear mapping $A \in {\mathcal {L}} (X_1, \ldots , X_N; Y)$attains its norm if there exists $(z_1 ,\ldots , z_N ) \in S_{X_1} \times \cdots \times S_{X_N}$ such that $\Vert A(z_1 , \ldots , z_N )\Vert =\Vert A\Vert$, where $\Vert A\Vert = \sup \Vert A(x_1, \ldots , x_N)\Vert$, the supremum being taken over all the elements $(x_1, \ldots , x_N) \in S_{X_1} \times \cdots \times S_{X_N}$. We denote by ${{\,\mathrm{NA}\,}}(X_1, \ldots , X_N; Y)$ the set of all norm attaining N-linear mappings.

Definition 1.1

[4] We say that $(X_1, \ldots , X_N; Y)$ has the Bishop–Phelps–Bollobás property for N-linear mappings (BPBp for N-linear mappings, for short) if given $\varepsilon > 0$, there exists $\eta (\varepsilon ) > 0$ such that whenever $A \in {\mathcal {L}} (X_1, \ldots , X_N; Y)$ with $\Vert A\Vert = 1$ and $\left( x_1, \ldots , x_N\right) \in S_{X_1} \times \cdots \times S_{X_N}$ satisfy

$$\begin{aligned} \left\| A\left( x_1 , \ldots , x_N \right) \right\| > 1 - \eta (\varepsilon ), \end{aligned}$$

there are $B \in {\mathcal {L}} (X_1, \ldots , X_N; Y)$ with $\Vert B\Vert = 1$ and $\left( z_1, \ldots , z_N\right) \in S_{X_1} \times \cdots \times S_{X_N}$ such that

$$\begin{aligned} \left\| B\left( z_1 , \ldots , z_N \right) \right\| = 1, \ \ \ \max _{1 \leqslant j \leqslant N} \Vert z_j - x_j \Vert< \varepsilon , \ \ \ \text{ and } \ \ \Vert B - A\Vert < \varepsilon . \end{aligned}$$

When $N = 1$, we simply say that the pair (X; Y) satisfies the BPBp (see [3, Definition 1.1]). Note that the Bishop–Phelps–Bollobás theorem asserts that the pair $(X; {\mathbb {K}})$ has the BPBp for every Banach space X. It is immediate to notice that if the pair (X; Y) has BPBp, then $\overline{{{\,\mathrm{NA}\,}}(X;Y)}={\mathcal {L}}(X;Y)$. However, the converse is not true even for finite dimensional spaces. Indeed, for a finite dimensional Banach space X, the fact that $B_X$ is compact implies that every bounded linear operator on X attains its norm, but it is known that there is some Banach space $Y_0$ so that the pair $(\ell _1^2; Y_0)$fails the BPBp (see [6, Example 4.1]). This shows that the study of the BPBp is not just a trivial extension of the density of norm attaining operators.

Similar to the case of operators, there was a lot of attention to the study of the denseness of norm attaining bilinear mappings. It was proved that, in general, there is no Bishop–Phelps theorem for bilinear mappings (see [2, Corollary 4]). Moreover, it is known that $\overline{{{\,\mathrm{NA}\,}}(^2 L^1[0,1];{\mathbb {K}})} \ne {\mathcal {L}}(^2 L^1[0,1];{\mathbb {K}})$ (see [11, Theorem 3]), even though $\overline{{{\,\mathrm{NA}\,}}(L^1[0,1];L_\infty [0,1])} = {\mathcal {L}}(L^1[0,1];L_\infty [0,1])$ (see [24]). This result is interesting since the Banach space ${\mathcal {L}}(X_1,X_2;{\mathbb {K}})$ is isometrically isomorphic to ${\mathcal {L}}(X_1;X_2^*)$ via the canonical isometry $A\in {\mathcal {L}}(X_1,X_2;{\mathbb {K}})\longmapsto T_A \in {\mathcal {L}}(X_1;X_2^*)$ given by $[T_A(x_1)](x_2) = A(x_1,x_2)$. Concerning the BPBp for bilinear mappings, it is known that $(\ell _1,\ell _1;{\mathbb {K}})$ fails the BPBp for bilinear mappings (see [12]) but the pair $(\ell _1, Y)$ satisfies the BPBp for many Banach spaces Y, including $\ell _\infty$ (see [3, Section 4]). We refer the papers [4, 16, 32] for more results on the BPBp for multilinear mappings.

Very recently, a stronger property than the BPBp was defined and studied.

Definition 1.2

[17] We say that $(X_1, \ldots , X_N; Y)$ has the Bishop–Phelps–Bollobás point property for N-linear mappings (BPBpp for N-linear mappings, for short) if given $\varepsilon > 0$, there exists $\eta (\varepsilon ) > 0$ such that whenever $A \in {\mathcal {L}} (X_1, \ldots , X_N; Y)$ with $\Vert A\Vert = 1$ and $\left( x_1, \ldots , x_N\right) \in S_{X_1} \times \cdots \times S_{X_N}$ satisfy

$$\begin{aligned} \left\| A\left( x_1 , \ldots , x_N \right) \right\| > 1 - \eta (\varepsilon ), \end{aligned}$$

there is $B \in {\mathcal {L}} (X_1, \ldots , X_N; Y)$ with $\Vert B\Vert = 1$ such that

$$\begin{aligned} \left\| B\left( x_1 , \ldots , x_N \right) \right\| = 1 \ \ \ \text{ and } \ \ \ \Vert B - A\Vert < \varepsilon . \end{aligned}$$

Clearly, the BPBpp implies the BPBp but the converse is not true in general. Actually, if the pair (X; Y) has the BPBpp for some Banach space Y, then X must be uniformly smooth (see [17, Proposition 2.3]). Also, it was proved in [17] that the pair $(X; {\mathbb {K}})$ has the BPBpp if and only if X is uniformly smooth. In both papers [17, 18] the authors presented differences between these two properties and found many positive examples having BPBpp.

On the other hand, one may think about a “dual” version of the BPBpp where, instead of fixing the point, we fix the operator.

Definition 1.3

[20] We say that $(X_1, \ldots , X_N; Y)$ has the Bishop–Phelps–Bollobás operator property for N-linear mappings (BPBop for N-linear mappings, for short) if given $\varepsilon > 0$, there exists $\eta (\varepsilon ) > 0$ such that whenever $A \in {\mathcal {L}} (X_1, \ldots , X_N; Y)$ with $\Vert A\Vert = 1$ and $\left( x_1, \ldots , x_N\right) \in S_{X_1} \times \cdots \times S_{X_N}$ satisfy

$$\begin{aligned} \left\| A\left( x_1 , \ldots , x_N \right) \right\| > 1 - \eta (\varepsilon ), \end{aligned}$$

there is $(z_1,\dots , z_N) \in S_{X_1}\times \cdots \times S_{X_N}$ such that

$$\begin{aligned} \left\| A\left( z_1 , \ldots , z_N \right) \right\| = 1 \ \ \ \text{ and } \ \ \ \max _{1 \leqslant j \leqslant N} \Vert x_j - z_j\Vert < \varepsilon . \end{aligned}$$

It was proved in [31] that the pair $(X; {\mathbb {K}})$ has the BPBop if and only X is uniformly convex. So, in the scalar-valued case, these two properties are dual from each other; that is, $(X; {\mathbb {K}})$ has the BPBpp if and only if $(X^*; {\mathbb {K}})$ has the BPBop. Nevertheless, it is known that there is no version for bounded linear operators of the BPBop. Indeed, in [19], it is proved that for $\dim (X), \dim (Y) \geqslant 2$, the pair (X; Y) always fails the BPBop. Hence, there is no hope for this “uniform” property, which lead us to consider a “local type” of it as in [15, 35, 36]. In these papers, the function $\eta$ in the definition of the BPBop depends not only on $\varepsilon$ but also on a fixed norm-one operator T, and some positive results are obtained, which are different from the uniform case when $\eta$ depends just on $\varepsilon$.

This motivated the authors of the present paper to study, in [20], all of the aforementioned properties in this local sense. In that paper, local versions of the BPBpp and BPBop (and also the BPBp) were addressed for linear operators. We give the precise definitions for N-linear mappings in Sect. 2. It turns out that these local properties are quite different from the corresponding uniform ones, as in the case of the BPBop (see [20, Section 5]). For instance, there is a close connection between those properties and the strongly subdifferentiability of the norm of the involved Banach spaces (see [20, Theorem 2.3]). Recall that a norm of X is said to be strongly subdifferentiable (SSD, for short) at a point $x \in S_X$ if the one-sided limit

$$\begin{aligned} \lim _{t \rightarrow 0^+} \frac{\Vert x + th\Vert - \Vert x\Vert }{t} \end{aligned}$$

(1)

exists uniformly for $h\in B_X$, and we say that the norm is SSD if it is SSD at every point in $S_X$. This differentiability is known to be strictly weaker than Fréchet differentiability. Let us notice that a norm $\Vert \cdot \Vert$ on a Banach space X is Fréchet differentiable at x if and only if $\Vert \cdot \Vert$ is both Gâteaux differentiable and SSD at x. Since this notion is a natural nonsmooth extension of Fréchet differentiability, it was considered by many authors in renorming theory. By the classical Dini theorem, it is known that every norm on a finite-dimensional Banach space is SSD at every point. It was proved in [22], as a particular case of a more general result which characterizes the points at which the sup-norm on $\ell _{\infty }$ is SSD, that the sup-norm on $c_0$ is SSD. This gives an example of a non-reflexive Banach space having a SSD norm. There are other examples of non-reflexive spaces X with a SSD norm, for instance, the predual of the Hardy space $H^1$ and the predual of the Lorentz space $L_{p,1}(\mu )$. Indeed $H^1$ and $L_{p,1}(\mu )$ are spaces with the $w^*$-Kadec–Klee property (see [21]) and this implies that the preduals have SSD norm (see [20, Proposition 2.6]). It is also known that if a norm of a Banach space X is SSD, then all of its separable subspaces have separable duals (see, for example, [27, Theorem 2] or [23, Theorem 5.1]). In other words, if X is SSD, then X is an Asplund space. For the reader who is interested in this topic, we suggest the already mentioned references [22, 23, 27] and also [14, 26].

Coming back to our “local BPBpp”, where $\eta$ depends on a point $x \in S_X$ and $\varepsilon > 0$, we have the following results.

Theorem 1.4

[20] Consider the following pairs of Banach spaces $(X; {\mathbb {K}})$ when X is

• (a) $c_0$ or

• (b) the predual of Lorentz sequence space $d_{*}(w, 1)$ or

• (c) the space VMO (which is the predual of the Hardy space $H^1$) or

• (d) a finite dimensional space,

and also the following pairs

• (e) $(\ell _1^N; L_p(\mu ))$ for $1< p < \infty , N \in {\mathbb {N}}$, and

• (f) $(c_0; L_p(\mu ))$ for $1 \leqslant p < \infty$.

Then, all of them satisfy this “local BPBpp”.

In this paper we continue the study of these local properties, emphasizing in the multilinear setting. Following the notation in [20], we use the symbol L$_{p, p}$ for the “local BPBpp”, when $\eta$ depends on a point $x \in S_X$ and L$_{o, o}$ for the “local BPBop”, when $\eta$ depends on an operator $T \in S_{{\mathcal {L}}(X, Y)}$ (see Definition 2.1 below). In the next section, we give the proper definitions and first results. Among others, we obtain the following results (see Proposition 2.3 and the comment below Corollary 2.5).

• If $(X_1, \ldots ,X_N;Y)$ has property L$_{p, p}$ (or L$_{o, o}$), then so does $(X_i;{\mathbb {K}})$ for every $1\leqslant i\leqslant N$.

• There exist (finite dimensional) Banach spaces $X_1,\dots , X_N, Y$ such that $(X_1, \ldots , X_N;Y)$ has the L$_{p, p}$ (respectively, L$_{o,o}$) but fails the BPBpp (respectively, BPBop).

We also focus on the bilinear case when the domains are $\ell _p$-spaces. In that sense, we obtain the following results (see Theorem 2.7 and Remark 2.9).

• If $2<p, q < \infty$, then $(\ell _p, \ell _q; {\mathbb {K}})$ has the L$_{p, p}$.

• If $1<p,q<\infty$, then $(\ell _p, \ell _q; {\mathbb {K}})$ has the L$_{o,o}$ if and only if $pq>p+q$. Hence, there exist spaces $\ell _p, \ell _q$ such that $(\ell _p,\ell _q;{\mathbb {K}})$ fails the bilinear L$_{o, o}$, while $(\ell _p;{\mathbb {K}})$ and $(\ell _q;{\mathbb {K}})$ have the linear L$_{o, o}$, since both are uniformly smooth.

In the proof of Theorem 2.7 we use a tensor product to prove that $(\ell _p, \ell _q; {\mathbb {K}})$ has the L$_{p, p}$ for $2<p, q < \infty$. As a consequence, we show that the norm of $\ell _p \hat{\otimes }_{\pi } \ell _q$ is strongly subdifferentiable for $2< p,q < \infty$. However if $p^{-1}+q^{-1} \geqslant 1$ or one of the indices p, q takes the value 1 or $\infty$, then its norm is not strongly subdifferentiable. In Sect. 3, motivated by the geometric property approximate hyperplane series property (AHSP, for short) in [3, 4], we get a characterization of strong subdifferentiability. The AHSP characterizes a Banach space Y for which $(\ell _1;Y)$ and $(\ell _1, Y;{\mathbb {K}})$ have the BPBp (in the linear and bilinear case, respectively). Although the pairs $(\ell _1;Y)$ and $(\ell _1, Y;{\mathbb {K}})$ do not have the L$_{p, p}$ (since $\ell _1$ is not SSD), we may ask if $(\ell _1^N;Y)$ and $(\ell _1^N, Y;{\mathbb {K}})$ have it. In Proposition 3.2 we prove a characterization for the norm of a Banach space Y to be strongly subdifferentiability. As a consequence of it, we prove that $(\ell _1^N, Y; {\mathbb {K}})$ has the L$_{p, p}$ for bilinear forms if and only if the norm of a Banach space Y is strongly subdifferentiable. Using similar ideas, we characterize the pairs $(\ell _1^N; Y)$ having the L$_{p, p}$ for operators, generalizing Theorem 1.4.(e). As a consequence of this last characterization, we prove that if a family $\{y_{\alpha } \}_{\alpha } \subset S_Y$ is uniformly strongly exposed with corresponding functionals $\{f_{\alpha }\}_{\alpha } \subset S_{Y^*}$, then $(\ell _1^N; Y)$ has the L$_{p, p}$ for operators whenever $\{f_{\alpha }\}_{\alpha }$ is a norming subset for the Banach space Y.

2 The L $_{p,p}$ and the L $_{o,o}$ for N-linear mappings

We start this section by giving the precise definitions of the local Bishop–Phelps–Bollobás properties for N-linear mappings. These are the analogous of [20, Definition 2.1].

Definition 2.1

(a) :

We say that $(X_1, \ldots , X_N; Y)$ has the L$_{p, p}$ if given $\varepsilon > 0$ and $(x_1,\dots ,x_N) \in S_{X_1} \times \cdots \times S_{X_N}$, there is $\eta (\varepsilon , (x_1,\dots ,x_N)) > 0$ such that whenever $A \in {\mathcal {L}}( X_1, \ldots , X_N; Y)$ with $\Vert A\Vert = 1$ satisfies

$$\begin{aligned} \Vert A(x_1,\dots ,x_N)\Vert > 1 - \eta (\varepsilon , (x_1,\ldots ,x_N)), \end{aligned}$$

there is $B \in {\mathcal {L}}(X_1,\ldots ,X_N;Y)$ with $\Vert B\Vert = 1$ such that

$$\begin{aligned} \Vert B(x_1,\dots ,x_N)\Vert = 1 \ \ \ \text{ and } \ \ \ \Vert B - A\Vert < \varepsilon . \end{aligned}$$

(b) :

We say that $(X_1,\ldots ,X_N;Y)$ has the L$_{o,o}$ if given $\varepsilon > 0$ and $A \in {\mathcal {L}}(X_1,\dots ,X_N;Y)$ with $\Vert A\Vert = 1$, there is $\eta (\varepsilon , A) > 0$ such that whenever $(x_1,\ldots ,x_N) \in S_{X_1} \times \cdots \times S_{X_N}$ satisfies

$$\begin{aligned} \Vert A(x_1,\dots ,x_N)\Vert > 1 - \eta (\varepsilon , A), \end{aligned}$$

there is $(z_1,\dots ,z_N) \in S_{X_1} \times \cdots \times S_{X_N}$ such that

$$\begin{aligned} \Vert A(z_1,\dots ,z_N)\Vert = 1 \ \ \text{ and } \ \ \max _{1 \leqslant j \leqslant N} \Vert x_j - z_j\Vert < \varepsilon . \end{aligned}$$

Let us observe that if $(X_1, \ldots , X_N; Y)$ satisfies the L$_{o,o}$, then every $A \in {\mathcal {L}}(X_1,\dots ,X_N;Y)$ attains its norm and, consequently, all the Banach spaces $X_i$’s must be reflexive. Indeed, if one of them is not reflexive, say $X_k$, by James theorem, there is $z_k^* \in S_{X_k^*}$ such that $|z_k^*(x_k)| < 1$ for all $x_k \in S_{X_k}$. Now, taking arbitrary $y_0\in S_Y$ and $z_i^* \in S_{X_i^*}$ for each $i\ne k$ and defining $A \in {\mathcal {L}}(X_1, \ldots ,X_N;Y)$ by $A(x_1,\ldots ,x_N) := \left( \Pi _{1\leqslant i \leqslant N} z_i^*(x_i)\right) y_0$, we see that A never attains its norm. Thus, in order to look for positive examples about the L$_{o,o}$, we must assume, at least, that $X_1, \ldots , X_N$ are all reflexive Banach spaces.

It was proved in [15, Theorem 2.4] that if X is a finite dimensional Banach space, then the pair (X; Y) has the L$_{o,o}$ for every Banach space Y. By using the similar proof, this can be generalized for N-linear mappings. However it does not hold for the L$_{p,p}$ in general. Indeed, suppose that Y is a strictly convex Banach space and that the pair $(\ell _1^2; Y)$ has the L$_{p,p}$. Then Y is uniformly convex by [20, Proposition 3.2]. So, choosing a strictly convex space $Y_0$ which is not uniformly convex, the pair $(\ell _1^2, Y_0)$ fails the L$_{p,p}$ although $\ell _1^2$ is 2-dimensional. In the case that Y is also finite dimensional, then we have a positive result as the following proposition. The proof is analogous to the operator case in [20, Proposition 2.8] and we omitted it.

Proposition 2.2

Let $N \in {\mathbb {N}}$ and let $X_1, \ldots , X_N$ be finite dimensional Banach spaces. Then,

• (a) $(X_1,\ldots ,X_N;Y)$ has the L$_{o,o}$ for every Banach space Y;

• (b) $(X_1,\ldots ,X_N;Y)$ has the L$_{p,p}$ for every finite dimensional Banach space Y.

It is known that if the pair (X; Y) satisfies the BPBpp or the BPBop or the L$_{p, p}$ for some Banach space Y, then so does $(X; {\mathbb {K}})$ (see [15, Proposition 2.9], [17, Proposition 2.7] and [20, Proposition 2.7], respectively). The same happens with property L$_{o, o}$. Indeed, given $\varepsilon > 0$ and $x^* \in S_{X^*}$, we construct, for a fixed $y_0 \in S_Y$, the operator $T \in {\mathcal {L}}(X, Y)$ given by $T(x) := x^*(x)y_0$ for all $x \in X$ and then we set $\eta (\varepsilon , x^*) := \eta (\varepsilon , T) > 0$. If $x_0 \in S_X$ is such that

$$\begin{aligned} |x^*(x_0)| > 1 - \eta (\varepsilon , x^*), \end{aligned}$$

then $\Vert T(x_0)\Vert > 1 - \eta (\varepsilon , T)$. Thus, there is $x_1 \in S_X$ such that

$$\begin{aligned} \Vert T(x_1)\Vert = |x^*(x_1)| = 1 \ \ \text{ and } \ \ \Vert x_1 - x_0\Vert < \varepsilon . \end{aligned}$$

Therefore, $(X, {\mathbb {K}})$ has the L$_{o, o}$. By using the same arguments, we can extend those results for N-linear mappings. In the proof, we use the canonical isometry between ${\mathcal {L}}(X_1, \ldots ,X_N;{\mathbb {K}})$ and ${\mathcal {L}}(X_1, \ldots ,X_{N-1};X_N^*)$ to deduce item (b) below.

Proposition 2.3

Let ${\mathcal {P}}$ be one of the properties BPBpp, BPBop, L$_{o,o}$ or L$_{p,p}$.

• (a) If $(X_1, \ldots ,X_N;Y)$ has the property ${\mathcal {P}}$, then so does $(X_1, \ldots ,X_N;{\mathbb {K}})$.

• (b) If $(X_1, \ldots ,X_N;{\mathbb {K}})$ has the property ${\mathcal {P}}$ and ${\mathcal {P}}$ is not L$_{p, p}$, then so does $(X_1, \ldots ,X_{N-1};X_N^*)$.

• (c) If $(X_1, \ldots ,X_N;Y)$ has the property ${\mathcal {P}}$, then so does $(X_i;{\mathbb {K}})$ for every $1\leqslant i\leqslant N$.

Proof

The proof of (a) and (b) is sketched above. To prove (c), it suffices to show that the pair $(X_1;{\mathbb {K}})$ has property ${\mathcal {P}}$ whenever $(X_1, \ldots ,X_N;Y)$ does. Suppose first that ${\mathcal {P}}$ is not L$_{p, p}$. Then, by item (a), we have that $(X_1, \ldots ,X_N;{\mathbb {K}})$ has property ${\mathcal {P}}$ and, in virtue of (b), $(X_1, \ldots ,X_{N-1};X_N^*)$ does. Applying (a) again, we see that $(X_1, \ldots ,X_{N-1};{\mathbb {K}})$ has property ${\mathcal {P}}$. That is, if $(X_1, \ldots ,X_N;{\mathbb {K}})$ has property ${\mathcal {P}}$, then $(X_1, \ldots ,X_{N-1};{\mathbb {K}})$ has property ${\mathcal {P}}$. Repeating this argument $(N-1)$-times, we see that $(X_1;{\mathbb {K}})$ has property ${\mathcal {P}}$.

Now, suppose that $(X_1, \ldots ,X_N;Y)$ has property L$_{p, p}$. Then, by (a), we have that $(X_1, \ldots ,X_N;{\mathbb {K}})$ has property L$_{p, p}$. Given $\varepsilon >0$ and $x_1^0\in S_{X_1}$, we want to see that there is $\eta (\varepsilon ,x_1^0)>0$ satisfying the definition of property L$_{p, p}$ for the pair $(X_1; {\mathbb {K}})$. Consider $(x_2^0,\dots ,x_N^0)\in S_{X_2}\times \cdots \times S_{X_N}$ and $(x_2^*,\ldots ,x_N^*)\in S_{X_2^*}\times \cdots \times S_{X_N^*}$ such that $x_i^*(x_i^0)=1$, for $i=2, \ldots , N$, and put $\eta (\varepsilon ,x_1^0):=\eta (\varepsilon ,(x_1^0,\ldots , x_N^0))$, which exists by hypothesis. Suppose that $x_1^*\in S_{X_1^*}$ is such that $|x_1^*(x_1^0)|>1-\eta (\varepsilon ,x_1^0)$. Then, defining $A(x_1,\dots , x_N)=x_1^*(x_1)x_2^*(x_2)\cdots x_N^*(x_N)$, we have that $A \in {\mathcal {L}}( X_1, \ldots , X_N; {\mathbb {K}})$, $\Vert A\Vert =1$, and

$$\begin{aligned} |A(x_1^0,\dots , x_N^0)|>1-\eta (\varepsilon ,(x_1^0,\ldots ,x_N^0)). \end{aligned}$$

Consequently, there exists $B \in {\mathcal {L}}(X_1, \ldots ,X_N;{\mathbb {K}})$ with $\Vert B\Vert = 1$ such that $|B(x_1^0, \ldots ,x_N^0)| = 1$ and $\Vert B - A\Vert < \varepsilon$. Therefore, defining $y_1^*\in X_1^*$ by $y_1^*(\cdot )=B(\ldots , x_2^0,\ldots , x_N^0)$, we see that

$$\begin{aligned} y_1^* \in S_{X_1^*}, \ \ \ |y_1^*(x_1^0)|=1, \ \ \ \text{ and } \ \ \ \Vert y_1^*-x_1^*\Vert \leqslant \Vert B - A\Vert < \varepsilon , \end{aligned}$$

which is the desired statement. $\square$

The item (b) above does not hold for the L$_{p, p}$; we provide a counterexample in Remark 3.4.

Due to a characterization of SSD due to Franchetti and Payá (see [23, Theorem 1.2]), we have that $(X; {\mathbb {K}})$ has the L$_{p,p}$ if and only if the norm of X is SSD and, by duality, $(X; {\mathbb {K}})$ has the L$_{o,o}$ if and only if X is reflexive and the norm of $X^*$ is SSD (see [20, Theorem 2.3] and also [27] where this fact was already observed).

By using this result and the characterization of property BPBpp for the pair $(X;{\mathbb {K}})$ given in [17, Proposition 2.1], we have the following consequences of Proposition 2.3.

Corollary 2.4

Let $N \in {\mathbb {N}}$ and $X_1, \ldots , X_N$ be Banach spaces.

• (a) If $(X_1, \ldots , X_N; Y)$ has the BPBpp for some Banach space Y, then $X_i$ is uniformly smooth for each $i=1,\ldots ,N$.

• (b) If $(X_1, \ldots , X_N;Y)$ has the L$_{p,p}$ for some Banach space Y, then $X_i$ is SSD for each $i=1,\ldots ,N$.

Another consequence of Proposition 2.3 is that, for spaces of dimension greater than 2, there is no BPBop for bilinear mappings. Indeed, if $\dim (X), \dim (Y) \geqslant 2$ and (X, Y; Z) has the BPBop for some Banach space Z, then by Proposition 2.3, the pair $(X, Y^*)$ has the BPBop for operators and, as we already mentioned in the Introduction, this is not possible. We can deduce the same for N-linear mappings.

Corollary 2.5

Let $N \in {\mathbb {N}}$. Let $X_i$ be a Banach space with $\dim (X_i) \geqslant 2$ for $1 \leqslant i \leqslant N$. Then, $(X_1, \ldots , X_N; Y)$ fails the BPBop for every Banach space Y.

At this point, we can point out some differences between properties BPBpp (respectively, BPBop) and L$_{p,p}$ (respectively, L$_{o,o}$). For instance, if $X_i=\ell _1^2$ or $\ell _\infty ^2$ and Y is any finite dimensional Banach space, then by Proposition 2.2 we have that $(X_1, \cdots , X_N;Y)$ has the L$_{p, p}$ (respectively, L$_{o,o}$) while, in virtue of Corollary 2.4.(a) (respectively, Corollary 2.5) it fails property BPBpp (respectively, BPBop).

Next we focus on the bilinear case when the domains are $\ell _p$-spaces. For the part (b) of Theorem 2.7 below we need the following lemma, which gives a converse of Proposition 2.3.(b) for property L$_{o, o}$.

Lemma 2.6

Let X, Y be Banach spaces and suppose that Y is uniformly convex. Then $(X, Y; {\mathbb {K}})$ has the L$_{o, o}$ for bilinear forms if and only if the pair $(X; Y^*)$ has the L$_{o, o}$ for operators.

Proof

From Proposition 2.3 (b), if $(X, Y; {\mathbb {K}})$ has the L$_{o, o}$ then so does $(X; Y^*)$. Hence, we only have to prove the converse. Let $\varepsilon \in (0, 1)$ be given. Since Y is uniformly convex, the pair $(Y;{\mathbb {K}})$ has the BPBop with some ${\tilde{\eta }}(\varepsilon ) > 0$ (see [31, Theorem 2.1]). This means that if $y^*\in S_{Y^*}$ and $y\in B_Y$ satisfy $|y^*(y)|>1-{\tilde{\eta }}(\varepsilon )$, then, there exists $z\in S_Y$ such that $|y^*(z)|=1$ and $\Vert y-z\Vert <\varepsilon$. Fix $A \in {\mathcal {L}}(X,Y; {\mathbb {K}})$ with $\Vert A\Vert = 1$ and take its associated operator $T_A \in S_{{\mathcal {L}}(X, Y^*)}$. Consider $\xi > 0$ to be such that $2\xi < \min \{{\tilde{\eta }}(\varepsilon ), \varepsilon \}$ and set

$$\begin{aligned} \eta (\varepsilon , A) := \min \{\xi , \eta '(\xi , T_A)\} > 0, \end{aligned}$$

where $\eta '$ is the function in the definition of L$_{o, o}$ for the pair $(X; Y^*)$. Let $(x_0, y_0) \in S_X \times S_Y$ be such that

$$\begin{aligned} |A(x_0, y_0)| > 1 - \eta (\varepsilon , A). \end{aligned}$$

Then, since

$$\begin{aligned} \Vert T_A(x_0)\Vert _{Y^*} \geqslant |T_A(x_0)(y_0)| = |A(x_0, y_0)| > 1 - \eta (\varepsilon , A) \geqslant 1 - \eta '(\xi , T_A), \end{aligned}$$

there is $x_1 \in S_X$ such that

$$\begin{aligned} \Vert T_A (x_1)\Vert _{Y^*} = 1 \ \ \ \text{ and } \ \ \ \Vert x_1 - x_0\Vert< \xi < \varepsilon . \end{aligned}$$

Now, since $T_A(x_1) \in S_{Y^*}$ and $y_0 \in S_Y$ satisfy

$$\begin{aligned} |[T_A(x_1)](y_0)|\geqslant & {} |T_A(x_0)(y_0)| - |T_A(x_1 - x_0)(y_0)| \\\geqslant & {} |A(x_0, y_0)| - \Vert x_1 - x_0\Vert \\> & {} 1 - \eta (\varepsilon , A) - \xi \\> & {} 1 - 2\xi > 1 - {\tilde{\eta }}(\varepsilon ), \end{aligned}$$

there is $y_1 \in S_Y$ such that $|[T_A(x_1)](y_1)| = 1$ and $\Vert y_1 - y_0\Vert < \varepsilon$. Since

$$\begin{aligned} 1 = |[T_A(x_1)](y_1)| = |A(x_1, y_1)|, \ \ \Vert x_1 - x_0\Vert< \varepsilon , \ \ \text{ and } \ \ \Vert y_1 - y_0\Vert < \varepsilon , \end{aligned}$$

we have proved that $(X,Y; {\mathbb {K}})$ has the L$_{o,o}$ for bilinear forms, as desired. $\square$

Denote by $X {{\hat{\otimes }}}_{\pi }Y$ the projective tensor product of the Banach spaces X and Y. Recall that the space ${\mathcal {L}}(X, Y; Z)$ is isometrically isomorphic to ${\mathcal {L}}( X {{\hat{\otimes }}}_{\pi }Y; Z)$ (see, for example, [34, Theorem 2.9]). Recall also the following definition: a dual Banach space $X^*$ has the $w^*$-Kadec–Klee property if $\Vert x_{\alpha }-x\Vert \rightarrow 0$ whenever $\Vert x_{\alpha }\Vert \rightarrow \Vert x\Vert$ and $x_\alpha \xrightarrow []{w^*}\, x$. If this holds for sequences, we say that $X^*$ has the sequential$w^*$-Kadec–Klee property. For some background concerning these properties, see [9, 28]. It is worth mentioning that if the unit ball $B_{X^*}$ is $w^*$-sequentially compact, then the sequential $w^*$-Kadec–Klee property implies the $w^*$-Kadec–Klee property on $X^*$ (see [9, Proposition 1.4]). Now, we prove the desired result.

Theorem 2.7

For $1<s<\infty$, let $s'$ be the conjugate of s (that is, $\frac{1}{s}+\frac{1}{s'}=1$).

• (a) If $2< p , q < \infty$, then $(\ell _p, \ell _q; {\mathbb {K}})$ has the L$_{p,p}$.

• (b) If $1<p,q<\infty$, then $(\ell _p, \ell _q; {\mathbb {K}})$ has the L$_{o,o}$ if and only if $pq>p+q$ (or, equivalently, $p>q'$).

Proof

(a) It is known that if $X^*$ has the $w^*$-Kadec–Klee property, then the pair $(X,{\mathbb {K}})$ has the L$_{p,p}$ (see [20, Proposition 2.6]). On the other hand, in [21, Theorem 4] it was proved that if $1<r<2<s<\infty$, then ${\mathcal {L}}(\ell _s; \ell _r)={\mathcal {L}}(\ell _s, \ell _{r'}; {\mathbb {K}}) = (\ell _s \hat{\otimes }_{\pi } \ell _{r'})^*$ has the sequential $w^*$-uniform-Kadec–Klee property, which implies the sequential $w^*$-Kadec–Klee property. Indeed, since $\ell _s \hat{\otimes }_{\pi } \ell _{r'}$ is reflexive (see, for instance, [34, Corollary 4.24]), then its unit dual ball is $w^*$-sequentially compact and, consequently, $(\ell _s \hat{\otimes }_{\pi } \ell _{r'})^*$ has the $w^*$-Kadec–Klee property. Hence, the pair $(\ell _p \hat{\otimes }_{\pi } \ell _{q};{\mathbb {K}})$ has the L$_{p,p}$ for $2< p, q < \infty$.

For a given $\varepsilon > 0$ and a fixed norm-one point $(x,y)\in S_{\ell _p}\times S_{\ell _q}$, consider $\eta (\varepsilon ,x\otimes y) > 0$ to be the function in the definition of L$_{p,p}$ for the pair $(\ell _p \hat{\otimes }_{\pi } \ell _{q};{\mathbb {K}})$. Let $A \in {\mathcal {L}}(\ell _p,\ell _q;{\mathbb {K}})$ with $\Vert A\Vert =1$ be such that

$$\begin{aligned} |A(x,y)| > 1 - \eta (\varepsilon , x \otimes y). \end{aligned}$$

Consider ${\hat{A}}$ to be the corresponding element in $S_{(\ell _p \hat{\otimes }_{\pi } \ell _{q})^*}$ via the canonical isometry. Then, we have

$$\begin{aligned} |{\hat{A}}(x\otimes y)|=|A(x,y)|>1-\eta (\varepsilon ,x\otimes y). \end{aligned}$$

Since the pair $(\ell _p \hat{\otimes }_{\pi } \ell _{q};{\mathbb {K}})$ has the L$_{p,p}$ with $\eta (\varepsilon , x \otimes y) > 0$, there exists ${\hat{B}}\in S_{(\ell _p \hat{\otimes }_{\pi } \ell _{q})^*}$ such that

$$\begin{aligned} |{\hat{B}}(x\otimes y)|=1 \ \ \ \text{ and } \ \ \ \Vert {\hat{B}}-{\hat{A}}\Vert <\varepsilon . \end{aligned}$$

Now we take $B\in S_{{\mathcal {L}}(\ell _p,\ell _q; {\mathbb {K}})}$, the corresponding element to ${\hat{B}}$ via the canonical isometry. Then, $|B(x, y)| = |{\hat{B}}(x\otimes y)|= 1$ and $\Vert B - A\Vert = \Vert {\hat{B}}-{\hat{A}}\Vert <\varepsilon$. This proves (a).

(b) Let $1<p,q<\infty$. By Lemma 2.6, $(\ell _p, \ell _q; {\mathbb {K}})$ has the L$_{o,o}$ if and only if $(\ell _p; \ell _{q'})$ has the L$_{o,o}$ and, in virtue of [15, Theorem 2.21], this happens if and only if $p>q'$. $\square$

Note that inside the proof of Theorem 2.7, we have proved that the pair $(\ell _p \hat{\otimes }_{\pi } \ell _{q};{\mathbb {K}})$ has the L$_{p,p}$ for $2< p, q < \infty$ for functionals. This yields to the following consequence.

Corollary 2.8

For $p,q\geqslant 1$

• (a) if $2<p, q < \infty$, then the norm of $\ell _p \hat{\otimes }_{\pi } \ell _{q}$ is SSD.

• (b) if $p^{-1}+q^{-1}\geqslant 1$ or one of p and q is 1 or $\infty$, then the norm of $\ell _p \hat{\otimes }_{\pi } \ell _{q}$ is not SSD.

Proof

As we already mentioned, item (a) follows from the proof of Theorem 2.7 and [20, Theorem 2.3]. To prove (b), note that if $p^{-1}+q^{-1}\geqslant 1$, then the main diagonal ${\mathcal {D}}=\overline{\text{ span } \{e_n\otimes e_n:\,\, n\in {\mathbb {N}}\}}$ is one-complemented in $\ell _p \hat{\otimes }_{\pi } \ell _{q}$ and isometrically isomorphic to $\ell _1$ (see, for instance, [5, Theorem 1.3]). Hence, if the norm of $\ell _p \hat{\otimes }_{\pi } \ell _{q}$ were SSD, by [20, Theorem 2.3] we would have that $(\ell _p \hat{\otimes }_{\pi } \ell _{q};{\mathbb {K}})$ has the L$_{p,p}$ and, by [20, Proposition 4.4 (b)], $(\ell _1; {\mathbb {K}})$ would have the L$_{p,p}$, which gives the desired contradiction. Suppose now that p or q take the value 1 or $\infty$. As we showed in the proof of Theorem 2.7, if $\ell _p \hat{\otimes }_{\pi } \ell _{q}$ were SSD then $(\ell _p, \ell _q; {\mathbb {K}})$ would have the L$_{p,p}$ for bilinear forms, which is not possible by Proposition 2.3.(c) since neither $\ell _1$ nor $\ell _\infty$ are not SSD [23, 27]. $\square$

In the proof of Theorem 2.7 we showed that if the pair $(\ell _p \hat{\otimes }_{\pi } \ell _{q};{\mathbb {K}})$ has the L$_{p,p}$ (or, equivalently, $\ell _p \hat{\otimes }_{\pi } \ell _{q}$ is SSD) then $(\ell _p, \ell _q; {\mathbb {K}})$ has the L$_{p,p}$ for bilinear forms. However, it is worth remarking that the converse is not true. For instance, $(\ell _2, \ell _2; {\mathbb {K}})$ has the L$_{p,p}$ for bilinear forms (moreover, it has the BPBpp by [17, Corollary 3.2]) but $\ell _2 \hat{\otimes }_{\pi } \ell _{2}$ is not SSD by using Corollary 2.8.(b).

We finish this section with some remarks and open questions.

Remark 2.9

(a) :

Since the uniform properties imply the local properties, when trying to prove that $(X_1,\dots , X_N;{\mathbb {K}})$ has the L$_{p,p}$ (respectively, L$_{o,o}$) for some Banach spaces $X_1,\dots , X_N$, it is natural to ask first if $(X_1,\dots , X_N;{\mathbb {K}})$ has (or not) the BPBpp (respectively, BPBop). Taking into account Theorem 2.7 we must say that, to the best of our knowledge, it is not known whether $(\ell _p, \ell _q; {\mathbb {K}})$ has the BPBpp when $2< p , q < \infty$. On the other hand, by Corollary 2.5, $(\ell _p, \ell _q; {\mathbb {K}})$ fails the BPBop for every $1<p, q<\infty$.

(b) :

By Proposition 2.3 we know that if $(X_1,\dots , X_N;{\mathbb {K}})$ has the L$_{p,p}$ (respectively, L$_{o,o}$), then so does $(X_i;{\mathbb {K}})$ for $1\leqslant i\leqslant N$. Hence, we may ask if $(X_1,\dots , X_N;{\mathbb {K}})$ has one of the mentioned properties whenever the pairs $(X_i;{\mathbb {K}})$ does. In that sense, note that $(\ell _p; {\mathbb {K}})$ and $(\ell _q;{\mathbb {K}})$ both have the L$_{o,o}$ for every $1<p, q<\infty$ (since $\ell _p$ and $\ell _q$ are both reflexive and $\ell _p^*, \ell _q^*$ are both SSD) but, in virtue of Theorem 2.7.(b), there are p, q such that $(\ell _p, \ell _q; {\mathbb {K}})$ fails the L$_{o,o}$. We also have that the pairs $(\ell _p;{\mathbb {K}})$ and $(\ell _q; {\mathbb {K}})$ have the L$_{p,p}$ for every $1<p, q<\infty$ but we do not know if there is some $1<p,q<\infty$ such that $(\ell _p, \ell _q; {\mathbb {K}})$ fails the L$_{p,p}$ for bilinear forms.

3 Local AHSP

Our main aim in this section is to give a characterization for the Banach space Y in such a way that $(\ell _1^N, Y; {\mathbb {K}})$ satisfies the L$_{p,p}$. Indeed, we prove that the norm of a Banach space Y is SSD if and only if $(\ell _1^N, Y; {\mathbb {K}})$ has the L$_{p, p}$ for bilinear forms. To do so, we get a characterization of SSD that is motivated by the approximate hyperplane series property (AHSP, for short), which was defined for the first time in [3]. Before giving our characterization, we recall the definition and important results concerning this property.

Definition 3.1

[3] A Banach space Y has the AHSP if for every $\varepsilon > 0$, there is $\eta (\varepsilon ) > 0$ such that given a sequence $(y_k) \subset S_Y$ and a convex series $\sum _{k=1}^{\infty } \alpha _k$ such that

$$\begin{aligned} \left\| \sum _{k=1}^{\infty } \alpha _k y_k \right\| > 1 - \eta (\varepsilon ), \end{aligned}$$

there exist $A \subset {\mathbb {N}}$, $y^* \in S_{Y^*}$, and $\{z_j: j \in A\}$ satisfying the following conditions:

$$\begin{aligned} \sum _{k \in A} \alpha _k > 1 - \varepsilon , \ \ \ \Vert z_k - y_k\Vert < \varepsilon , \ \ \ \text{ and } \ \ \ y^*(z_k) = 1 \text { for every } k \in A. \end{aligned}$$

Finite dimensional, uniformly convex and lush spaces are known examples of Banach spaces satisfying the AHSP (see [3, Propositions 3.5, 3.8] and [13, Theorem 7], respectively). More specifically, $L_p(\mu )$-spaces for arbitrary $1 \leqslant p \leqslant \infty$ and C(K)-spaces for a compact Hausdorff K are concrete examples of such a Banach spaces. This property was defined in [3] in order to give a characterization for the Banach spaces Y such that the pair $(\ell _1; Y)$ has the BPBp for operators. Here, we are interested to get a local version of AHSP which is related with the L$_{p,p}$ for bilinear mappings (see [4, Definition 3.1] and [16, Section 3] for AHSP for bilinear mappings). It turns out that this local version of AHSP is equivalent to SSD of the norm.

Proposition 3.2

Let Y be a Banach space. For any $N \in {\mathbb {N}}$, the following are equivalent.

• (a) The norm of Y is SSD.

• (b) Given $\varepsilon > 0$, a nonempty set $A \subset \{1, \ldots , N \}$, $(\alpha _j)_{j \in A}$ with $\alpha _j > 0$ for all $j \in A$ and $\sum _{j \in A} \alpha _j = 1$, and $y \in S_Y$, there is $\eta = \eta (\varepsilon , (\alpha _j)_{j \in A}, y) > 0$ such that whenever $(y_j^*)_{j \in A} \subset S_{Y^*}$ satisfies

  $$\begin{aligned} {{\,\mathrm{Re}\,}}\sum _{j \in A} \alpha _j y_j^*(y) > 1 - \eta , \end{aligned}$$

  there is $(z_j^*)_{j \in A} \subset S_{Y^*}$ such that

  $$\begin{aligned} z_j^*(y) = 1 \ \ \ \text{ and } \ \ \ \Vert z_j^* - y_j^*\Vert < \varepsilon , \end{aligned}$$

  for all $j \in A$.

Proof

(b) implies (a) by considering a singleton A and recalling that Y is SSD if and only if $(Y,{\mathbb {K}})$ has the L$_{p,p}$. Now assume that the norm of Y is SSD or, equivalently, that the pair $(Y; {\mathbb {K}})$ has the L$_{p,p}$ with some function $\eta '$. Fix $\varepsilon > 0$, a nonempty set $A \subset \{1, \ldots , N \}$, $(\alpha _j)_{j \in A}$ with $\alpha _j > 0$ for all $j \in A$ and $\sum _{j \in A} \alpha _j = 1$, and $y \in S_Y$. Set $\alpha := \min _{j\in A}\alpha _j$ and

$$\begin{aligned} \eta (\varepsilon , (\alpha _j)_{j \in A}, y) := \alpha \eta '\left( \frac{\varepsilon ^2}{16},y \right) > 0. \end{aligned}$$

Note that we may assume that $\eta '(\varepsilon , y) \leqslant \varepsilon$ for every $\varepsilon > 0$. Let $(y_j^*)_{j \in A} \subset S_{Y^*}$ be such that

$$\begin{aligned} {{\,\mathrm{Re}\,}}\sum _{j \in A} \alpha _j y_j^*(y) > 1 - \eta (\varepsilon , (\alpha _j)_{j \in A}, y). \end{aligned}$$

Then, for each $k \in A$, we have

$$\begin{aligned} 1 - \alpha _k \eta '(\varepsilon , y)\leqslant & {} 1 - \alpha \eta '(\varepsilon , y) = 1 - \eta (\varepsilon , (\alpha _j)_{j \in A}, y) \\< & {} {{\,\mathrm{Re}\,}}\sum _{j \in A} \alpha _j y_j^*(y) \leqslant {{\,\mathrm{Re}\,}}\alpha _k y_k^*(y) + (1 - \alpha _k). \end{aligned}$$

So,

$$\begin{aligned} {{\,\mathrm{Re}\,}}y_k^*(y) > 1 - \eta '\left( \frac{\varepsilon ^2}{16},y \right) \ \ \text{ for } \text{ each } k \in A. \end{aligned}$$

Since $(Y; {\mathbb {K}})$ has the L$_{p,p}$ with $\eta '$, for each $k \in A$, there is ${\tilde{z}}_k^* \in S_{Y^*}$ such that

$$\begin{aligned} |{\tilde{z}}_k^*(y)| = 1 \ \ \ \text{ and } \ \ \ \Vert {\tilde{z}}_k^* - y_k^*\Vert < \frac{\varepsilon ^2}{16}. \end{aligned}$$

For each $k \in A$, write ${\tilde{z}}_k^*(y) = e^{i \theta _k} |{\tilde{z}}_k^*(y)| = e^{i \theta _k}$. Then,

$$\begin{aligned} \Vert e^{-i\theta _k} {\tilde{z}}_k^* - y_k^*\Vert \leqslant |1 - e^{i \theta _k}| + \Vert {\tilde{z}}_k^* - y_k^*\Vert \end{aligned}$$

for all $k \in A$. Now, note that whenever $k \in A$, we have

$$\begin{aligned} {{\,\mathrm{Re}\,}}e^{i \theta _k} = {{\,\mathrm{Re}\,}}{\tilde{z}}_k^*(y) \geqslant {{\,\mathrm{Re}\,}}y_k^*(y) - \Vert {\tilde{z}}_k^* - y_k^*\Vert> 1 - \eta '\left( \frac{\varepsilon ^2}{16},y \right) - \frac{\varepsilon ^2}{16} > 1 - \frac{\varepsilon ^2}{8}. \end{aligned}$$

So,

$$\begin{aligned} |1 - e^{i \theta _k}|^2= & {} ({{\,\mathrm{Re}\,}}e^{i \theta _k} - 1)^2 + ({{\,\mathrm{Im}\,}}e^{i \theta _k} )^2 \\= & {} 1 - 2 {{\,\mathrm{Re}\,}}e^{i \theta _k} +({{\,\mathrm{Re}\,}}e^{i \theta _k})^2 + ({{\,\mathrm{Im}\,}}e^{i \theta _k} )^2 \\= & {} 2 (1 - {{\,\mathrm{Re}\,}}e^{i \theta _k}) < \frac{\varepsilon ^2}{4}, \end{aligned}$$

which implies $|1 - e^{i \theta _k}| < \frac{\varepsilon }{2}$ for every $k \in A$. Then, for each $k \in A$, we have

$$\begin{aligned} \Vert e^{-i\theta _k}{\tilde{z}}_k^* - y_k^*\Vert< \frac{\varepsilon }{2} + \frac{\varepsilon ^2}{16} < \varepsilon . \end{aligned}$$

Setting $z_k^* := e^{-i \theta _k} {\tilde{z}}_k^*$ for each $k \in A$, we have that $z_k^*(y) = 1$ and $\Vert z_k^* - y_k^*\Vert < \varepsilon$, which proves that (a) implies (b). $\square$

Note that part (b) of Proposition 3.2 is a kind of local version of AHSP for the Bishop–Phelps–Bollobás point property since we do not move the initial point and also the $\eta$ in its definition depends not only on a positive $\varepsilon$ but also on a finite convex series and on a norm-one point. Observe that, by a simple change of parameters, we can take $(y_j^*)_{j \in A}$ in $B_{Y^*}$ instead of $S_{Y^*}$ in item (b) and we are using this fact without any explicit reference in the next theorem, where we prove the promised characterization of property L$_{p,p}$ for $(\ell _1^N, Y; {\mathbb {K}})$.

Theorem 3.3

Let Y be a Banach space and $N \in {\mathbb {N}}$. Then, $(\ell _1^N, Y;{\mathbb {K}})$ has the L$_{p, p}$ if and only if the norm of Y is SSD.

Proof

If $(\ell _1^N, Y; {\mathbb {K}})$ has the L$_{p, p}$ for bilinear forms then, by Proposition 2.3.(c), the pair $(Y; {\mathbb {K}})$ has the L$_{p, p}$, which is equivalent to say that the norm of Y is SSD. Suppose now that the norm of Y is SSD. Let $\varepsilon > 0$ and $(x, y) \in S_{\ell _1^N} \times S_Y$ be given. We write $x = (x_1, \ldots , x_N)$ and assume that $x_j \geqslant 0$ for all $j = 1, \ldots , N$ by composing it with an isometry if necessary. Let $A = \{j \in \{1, \ldots , N\}: x_j \not = 0 \}$. Then $\Vert x\Vert _1 = \sum _{j \in A} x_j = 1$. Consider $(x_j)_{j \in A}$ and by Proposition 3.2, we may set

$$\begin{aligned} \eta (\varepsilon , (x, y)) := \eta (\varepsilon , (x_j)_{j \in A}, y) > 0. \end{aligned}$$

Let $A \in {\mathcal {L}}(\ell _1^N, Y; {\mathbb {K}})$ with $\Vert A\Vert = 1$ be such that

$$\begin{aligned} |A(x, y)| > 1 - \eta (\varepsilon , (x, y)). \end{aligned}$$

By rotating A, if necessary, we may assume that ${{\,\mathrm{Re}\,}}A (x, y) > 1 - \eta (\varepsilon , (x, y))$. So,

$$\begin{aligned} {{\,\mathrm{Re}\,}}A(x, y) = {{\,\mathrm{Re}\,}}\sum _{j \in A} x_j A(e_j, y) > 1 - \eta (\varepsilon , (x, y)). \end{aligned}$$

Define $y_j^* (y) := A(e_j, y)$ for every $y \in Y$. Since $\Vert A\Vert = 1$, we have that $(y_j^*)_{j \in A} \subset B_{Y^*}$ and

$$\begin{aligned} {{\,\mathrm{Re}\,}}\sum _{j \in A} x_j y_j^*(y) > 1 - \eta (\varepsilon , (x, y)) = 1 - \eta (\varepsilon , (x_j)_{j \in A}, y). \end{aligned}$$

By Proposition 3.2, there is $(z_j^*)_{j \in A} \subset S_{Y^*}$ such that $z_j^*(y) = 1$ and $\Vert z_j^* - y_j^*\Vert < \varepsilon$, for all $j \in A$. Now, define $B: \ell _1^N \times Y \longrightarrow {\mathbb {K}}$ by

$$\begin{aligned} B(u, v) := \sum _{j \in A} u_j z_j^*(v) + \sum _{j \not \in A} u_j A(e_j, v), \end{aligned}$$

for $u = (u_j)_{j=1}^N \in \ell _1^N$ and $v \in Y$. So, $\Vert B\Vert \leqslant 1$ and

$$\begin{aligned} |B(x, y)| = \left| \sum _{j \in A} x_j B(e_j, y) \right| = \left| \sum _{j \in A} x_j z_j^*(y) \right| = \sum _{j \in A} x_j = 1. \end{aligned}$$

Then $\Vert B\Vert = |B(x, y)| = 1$. Also, for every $(u, v) \in S_{\ell _1^N} \times S_Y$, we have

$$\begin{aligned} |B(u, v) - A(u, v)| = \left| \sum _{j \in A} u_j (z_j^* - y_j^*)(v) \right| < \varepsilon \sum _{j \in A} u_j \Vert v\Vert \leqslant \varepsilon . \end{aligned}$$

Therefore $\Vert B - A\Vert < \varepsilon$, and this shows that $(\ell _1^N, Y; {\mathbb {K}})$ has the L$_{p, p}$ for bilinear forms. $\square$

Remark 3.4

We can use Theorem 3.3 to show that Proposition 2.3.(b) does not hold for L$_{p, p}$. Consider a dual space $Y^*$ which is isomorphic to $\ell _1$ and its norm is locally uniformly rotund (and then strictly convex) [29, 30]. Then, the norm of Y is Fréchet differentiable (see, for example, [25, Fact 8.18]), and so it is SSD. By Theorem 3.3, we have that $(\ell _1^N,Y;{\mathbb {K}})$ has the L$_{p, p}$. Suppose by contradiction that the pair $(\ell _1^N; Y^*)$ has the L$_{p, p}$ for operators. Since L$_{p, p}$ is stable under one-complemented subspaces on the domain (see [20, Proposition 4.4]), the pair $(\ell _1^2; Y^*)$ also has the L$_{p, p}$. Since $Y^*$ is strictly convex, by using [20, Proposition 3.2] we get that $Y^*$ should be uniformly convex, which is not possible. So, Y is the desired counterexample.

Analogously to the bilinear case, we obtain a characterization of those Banach spaces Y such that the pair $(\ell _1^N; Y)$ has the L$_{p, p}$ for operators for an arbitrary $N \in {\mathbb {N}}$. Since the proof is quite similar to Theorem 3.3, we omit the details.

Proposition 3.5

Let Y be a Banach space. The pair $(\ell _1^N; Y)$ has the L$_{p, p}$ for operators if and only if given $\varepsilon >0$, a nonempty set $A \subset \{1, \ldots , N \}$ and $(\alpha _j)_{j \in A}$ with $\alpha _j > 0$ for all $j \in A$ such that $\sum _{j \in A} \alpha _j = 1$, there is $\eta = \eta (\varepsilon , (\alpha _j)_{j \in A}) > 0$ such that whenever $(y_j)_{j \in A} \subset S_Y$ satisfies

$$\begin{aligned} \left\| \sum _{j \in A} \alpha _j y_j \right\| > 1 - \eta , \end{aligned}$$

there are $z^* \in S_{Y^*}$ and $(z_j)_{j \in A} \subset S_Y$ such that

$$\begin{aligned} z^*(z_j) = 1 \ \ \ \text{ and } \ \ \ \Vert z_j - y_j\Vert < \varepsilon , \end{aligned}$$

for all $j \in A$.

It turns out that the AHSP (see Definition 3.1) implies the characterization of Proposition 3.5, as we show in the following theorem. This provides new examples of spaces Y such that $(\ell _1^N; Y)$ has the L$_{p, p}$ for linear operators. In particular, if Y is a uniformly convex Banach space, then the pair $(\ell _1^N; Y)$ has the L$_{p, p}$, a result that was already proved in [20, Proposition 2.10].

Theorem 3.6

Let Y be a Banach space and $N \in {\mathbb {N}}$. If Y has AHSP, then $(\ell _1^N; Y)$ has the L$_{p, p}$.

Proof

Assume that Y has AHSP with a function $\eta$ and fix $\varepsilon >0$, a nonempty $A \subset \{1, \ldots , N \}$ and a sequence of positive numbers $(\alpha _j)_{j \in A}$ with $\sum _{j \in A} \alpha _j = 1$. Take $0<\lambda <\min \big \{\varepsilon ,\min \{\alpha _j~:~j\in A\}\big \}$ and assume that a sequence of vectors $(y_j)_{j\in A}\subset S_Y$ satisfies

$$\begin{aligned} \left\| \sum _{j \in A} \alpha _j y_j \right\| > 1 - \eta (\lambda ). \end{aligned}$$

By the definition of AHSP, there are $B\subset A$, $\{z_k : k\in B\}\subset S_X$ and $z^*\in S_{Y^*}$ such that

$$\begin{aligned} \sum _{k\in B} \alpha _k>1-\lambda , \ \ \ \Vert z_k-x_k\Vert <\lambda , \ \ \ \text{ and } \ \ \ x^*(z_k)=1 \end{aligned}$$

for all $k\in B$. Since $\lambda <\min \{\alpha _j~:~j\in A\}$, $A=B$. By Proposition 3.5, $(\ell _1^N; Y)$ has the L$_{p, p}$. $\square$

Corollary 3.7

Let Y be a Banach space and $N \in {\mathbb {N}}$. If Y is a

• (a) finite dimensional space or

• (b) uniformly convex space or

• (c) lush space,

then the pair $(\ell _1^N; Y)$ has the L$_{p, p}$ for operators.

We also get a result about uniformly strongly exposed family. We say that a family $\{y_{\alpha }\}_{\alpha } \subset S_Y$ is uniformly strongly exposed with respect to a family $\left\{ f_\alpha \right\} _\alpha \subset S_{Y^*}$, if there is a function $\varepsilon \in (0,1) \mapsto \delta (\varepsilon )>0$ such that $f_\alpha (y_\alpha ) = 1$ for all $\alpha$ and ${{\,\mathrm{Re}\,}}f_\alpha (y)> 1-\delta (\varepsilon )$ implies $\Vert y-y_\alpha \Vert <\varepsilon$ whenever $y\in B_X$.

Proposition 3.8

Let Y be a Banach space and let $\{y_{\alpha }\}_{\alpha } \subset S_Y$ be a uniformly strongly exposed family with corresponding functionals $\{f_{\alpha }\}_{\alpha } \subset S_{Y^*}$. If $\{f_{\alpha }\}_{\alpha }$ is a norming subset for Y, then the pair $(\ell _1^N; Y)$ has the L$_{p, p}$.

Proof

Let $N \in {\mathbb {N}}$ and $A \subset \{1, \ldots , N\}$ be a nonempty finite subset. Let $\varepsilon > 0$ and suppose there is $\alpha := (\alpha _{j})_{j \in A}$ such that $\alpha _j > 0$ for all $j \in A$ and $\sum _{j \in A} \alpha _j = 1$. Set $K_{\alpha } := \min \{ \alpha _j: j \in A \}$ and define

$$\begin{aligned} \eta = \eta (\varepsilon , (\alpha _j)_{j \in A}) := K_{\alpha } \delta \left( \frac{\varepsilon }{2} \right) > 0, \end{aligned}$$

where $\varepsilon \mapsto \delta (\varepsilon )$ is the function for the family $\{y_{\alpha }\}_{\alpha }$. Let $(y_j)_{j \in A} \subset S_Y$ be such that

$$\begin{aligned} \left\| \sum _{j \in A} \alpha _j y_j \right\| > 1 - \eta . \end{aligned}$$

Since $\{f_{\alpha }\}$ is norming for Y, we may take $\alpha _0$ to be such that

$$\begin{aligned} \sum _{j \in A} \alpha _j {{\,\mathrm{Re}\,}}f_{\alpha _0} (y_j) = {{\,\mathrm{Re}\,}}f_{\alpha _0} \left( \sum _{j \in A} \alpha _j y_j \right) > 1 - \eta . \end{aligned}$$

Then, for each $i \in A$, we have

$$\begin{aligned}&1 - K_{\alpha } \delta \left( \frac{\varepsilon }{2}\right) < \sum _{j \in A} \alpha _j {{\,\mathrm{Re}\,}}f_{\alpha _0} (y_j) \leqslant \sum _{j \in A \setminus \{i\}} \alpha _j + \alpha _i {{\,\mathrm{Re}\,}}f_{\alpha _0} (y_i)\\&\quad =1 - \alpha _i + \alpha _i {{\,\mathrm{Re}\,}}f_{\alpha _0}(y_i). \end{aligned}$$

Therefore, for each $i \in A$, we have

$$\begin{aligned} 1 - {{\,\mathrm{Re}\,}}f_{\alpha _0} (y_i) < \frac{K_{\alpha }}{\alpha _i} \delta \left( \frac{\varepsilon }{2} \right) \leqslant \delta \left( \frac{\varepsilon }{2} \right) , \end{aligned}$$

which implies that ${{\,\mathrm{Re}\,}}f_{\alpha _0} (y_i) > 1 - \left( \frac{\varepsilon }{2} \right)$ for every $i \in A$. So, we have that $\Vert y_i - y_{\alpha _0}\Vert < \frac{\varepsilon }{2}$ for all $i \in A$. Thus, for every $n, m \in A$, we have that

$$\begin{aligned} \Vert y_n - y_m\Vert \leqslant \Vert y_n - y_{\alpha _0}\Vert + \Vert y_{\alpha _0} - y_m\Vert < \varepsilon . \end{aligned}$$

Since $A \not = \emptyset$, choose $n_0 \in A$ and set $z_j := y_{n_0}$ for all $j \in A$. Then, $z_j \in S_Y$ and $\Vert z_j - y_j\Vert = \Vert y_{n_0} - y_j\Vert < \varepsilon$ for all $j \in A$. Finally, take $y^* \in S_{Y^*}$ to satisfy $y^*(z_j) = y^*(y_{n_0}) = 1$. By Proposition 3.5, $(\ell _1^N; Y)$ has the L$_{p, p}$. $\square$