SOME REMARKS ON THE WEAK MAXIMIZING PROPERTY

SHELDON DANTAS, MINGU JUNG, AND GONZALO MARTINEZ-CERVANTES

Abstract. A pair of Banach spaces (E,F ) is said to have the weak maximizing prop-erty (WMP, for short) if for every bounded linear operator T from E into F , the existenceof a non-weakly null maximizing sequence for T implies that T attains its norm. Thisproperty was recently introduced in an article by R. Aron, D. Garcıa, D. Pelegrino andE. Teixeira, raising several open questions. The aim of the present paper is to con-tribute to the better knowledge of the WMP and its limitations. Namely, we providesufficient conditions for a pair of Banach spaces to fail the WMP and study the be-haviour of this property with respect to quotients, subspaces, and direct sums, whichopen the gate to present several consequences. For instance, we deal with pairs of the form(Lp[0, 1], Lq[0, 1]), proving that these pairs fail the WMP whenever p > 2 or q < 2. Wealso show that, under certain conditions on E, the assumption that (E,F ) has the WMPfor every Banach space F implies that E must be finite dimensional. On the other hand,we show that (E,F ) has the WMP for every reflexive space E if and only if F has theSchur property. We also give a complete characterization for the pairs (`s⊕p `p, `s⊕q `q)to have the WMP by calculating the moduli of asymptotic uniform convexity of `s⊕p `pand of asymptotic uniform smoothness of `s ⊕q `q when 1 < p 6 s 6 q < ∞. We con-clude the paper by discussing some variants of the WMP and presenting a list of openproblems on the topic of the paper.

1. Introduction

Given two Banach spaces E and F , the pair (E,F ) is said to satisfy the weak maxi-mizing property (WMP, for short) if a bounded linear operator T from E into F is norm-attaining whenever there exists a non-weakly null maximizing sequence for T . WhenE = F , we simply say that E satisfies the WMP. By a maximizing sequence for abounded linear operator T we mean a sequence (xn)n∈N of norm one elements such that‖T (xn)‖ −→ ‖T‖. This property was recently defined and studied in [2], motivated by[14, Theorem 1] which asserts that the pair (`p, `q) has the WMP for any 1 < p <∞ and1 6 q < ∞. This result provides a strong connection between the WMP and the theoryof norm-attaining operators. Indeed, it is observed by [2, Proposition 2.4] that if the pair(E,F ) has the WMP, then an operator T + K attains its norm whenever T ∈ L(E,F )and K ∈ K(E,F ) satisfies ‖T‖ < ‖T +K‖, generalizing a result due to J. Kover (see [9,Theorem 4]). As a consequence, the pair (E,F ) has the WMP for some non-trivial space

2010 Mathematics Subject Classification. Primary 46B20; Secondary 46B25, 46B28.Key words and phrases. Maximizing sequence; norm-attaining operators; James theorem.S. Dantas was supported by the project OPVVV CAAS CZ.02.1.01/0.0/0.0/16 019/0000778 and by

the Estonian Research Council grant PRG877. The second author was supported by NRF (NRF-2019R1A2C1003857). The third author was partially supported by Fundacion Seneca [20797/PI/18],Agencia Estatal de Investigacion [MTM2017-86182-P, cofunded by ERDF, EU] and by the EuropeanSocial Fund (ESF) and the Youth European Initiative (YEI) under Fundacion Seneca [21319/PDGI/19].

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2 DANTAS, JUNG, AND MARTINEZ-CERVANTES

F if and only if every bounded linear functional on E is norm-attaining (or, equivalently,that E is reflexive by using the classical James theorem), which implies that this newproperty can be viewed as an extended notion of reflexivity.

Even more recent is the paper [4], where the authors have provided a new plan of actionto deal with the WMP (and its variants) by using the modulus of asymptotic uniformconvexity and the modulus of asymptotic uniform smoothness (see [4, Theorem 3.1]). Inparticular, using these tools they manage to cover all the positive known examples from[2] and, furthermore, to present a big array of new pairs of Banach spaces satisfying theWMP. Namely, they deal with concrete Banach spaces such as James, Tsirelson and Orliczspaces.

In this paper, we present a natural family of pairs of Banach spaces which lack theWMP (see Theorem 2.1). These, together with the behaviour of the WMP with respectto quotients and subspaces (see Proposition 2.2), provide sufficient conditions for a pairof Banach spaces to fail the WMP. Most of the results in this paper follow from a suitablecombination of these two results in different settings. In particular, these results showthat the WMP is a geometric condition, not invariant under isomorphisms, much morerestrictive than reflexivity (see Corollary 2.3). Moreover, we deal with Lp-spaces and givepartial answers to a question posed in [2] (see Theorem 2.4 and Question 4.2): indeed,we prove that (Lp[0, 1], Lq[0, 1]) fails to have the WMP whenever p > 2 or q < 2. We alsoshow that, given a Banach space E satisfying certain natural conditions, if (E,F ) hasthe WMP for every Banach space F then E is finite-dimensional (see Proposition 2.6).On the other hand, it is immediate that every operator from a reflexive Banach spaceinto a Banach space with the Schur property is compact and, therefore, attains its norm.Thus, if F has the Schur property, then (E,F ) has the WMP for every reflexive Banachspace E (see [4, Proposition 5.2]). In Theorem 2.7 it is shown that, indeed, the Schurproperty is characterized by this property, i.e. that a Banach space F satisfies the Schurproperty if and only if (E,F ) has the WMP for every reflexive space E. Motivated by [9,Corollary 7], [2, Proposition 2.4], and a question posed by R. Aron in the Conference onFunction Theory on Infinite Dimensional Spaces XVI held in Madrid in 2019, we showin Proposition 2.8 that the inequality ‖T + K‖ > ‖T‖ implies that T + K attains thenorm whenever T ∈ L(E, c0) and K ∈ K(E, c0). Since there are pairs of Banach spaces(E, c0), even with E being reflexive, without the WMP, this shows that the converse of [2,Proposition 2.4] does not hold. We do not know if the converse might hold for pairs (E,F )with both spaces E and F being reflexive (see Question 4.4). Next, we provide a completecharacterization for the pair (`s⊕p `p, `s⊕q `q) to have the WMP by calculating the moduliof asymptotic uniform convexity of `s⊕p `p and asymptotic uniform smoothness of `s⊕q `qwhen 1 < p 6 s 6 q < ∞ (their formal definitions are given just after Theorem 2.11).Finally, we discuss some variants of the WMP and the relation among them (Proposition2.16), and devote Section 4 to present a list of open problems on the topic.

2. The Results

Let us start this section with the basic notation and definition we will be using through-out the paper. Let E and F be Banach spaces over a scalar field K, which can be the realnumbers R or the complex numbers C. Let BE and SE be the closed unit ball and the

SOME REMARKS ON THE WEAK MAXIMIZING PROPERTY 3

unit sphere of the Banach space E, respectively. We denote the set of all bounded linearoperators from E into F by L(E,F ). When E = F , we simply write L(E). K(E,F )stands for the set of compact operators. We say that T ∈ L(E,F ) is norm-attaining orattains its norm if there exists x0 ∈ SE such that ‖T (x0)‖ = ‖T‖ = supx∈BE

‖T (x)‖. Wedenote by NA(E,F ) the set of all norm-attaining operators from E into F . As mentionedearlier, a Banach space E must be reflexive whenever a pair of Banach spaces (E,F )satisfies the WMP for some non-trivial F by [2, Corollary 2.5]. This means that, in orderto look up for new examples of pairs (E,F ) satisfying such a property, we always have toassume previously that E is reflexive.

As a counterpart for the already mentioned paper [4], where some conditions for a pairof Banach spaces to have the WMP are given, the next theorem provides natural pairsof Banach spaces failing such a property. By a trivial Banach space we refer to a Banachspace consisting of just the vector zero.

Theorem 2.1. Let E,F,X, Y be nontrivial Banach spaces and suppose that there existsT 6∈ NA(E,F ). Then,

(a) the pair (E ⊕p X,F ⊕q Y ) fails the WMP whenever 1 6 q < p <∞;(b) the pair (E ⊕∞ X,F ) fails the WMP.

Before proving Theorem 2.1 (for its proof, see Section 3), let us point out some of itsrelevant consequences that we will be getting throughout the paper. In particular, weanswer [2, Questions 3.2], where the authors asked whether the WMP is preserved underisomorphisms and conjectured that the WMP for E is not equivalent to the fact that Eis reflexive (see Corollary 2.3). Also, we will use Theorem 2.1 as a tool to deal with pairsof the form (Lp[0, 1], Lq[0, 1]) and (`s ⊕p `p, `s ⊕q `q).

In order to provide such consequences, we will need the following elementary facts aboutthe behavior of the WMP when it comes to closed subspaces, direct sums, and quotientspaces.

In what follows, the symbol 6 stands for closed subspaces.

Proposition 2.2. Let E,F be Banach spaces.

(a) If (E,F ) has the WMP, then so does (E,F1) for every subspace F1 6 F .(b) If (E,F ) has the WMP, then so does (E/E1, F ) for every E1 6 E. In particular,

(E1, F ) has the WMP whenever E1 6 E is 1-complemented.(c) If both (E1, F ) and (E2, F ) have the WMP, then so does (E1 ⊕1 E2, F ).(d) If both (E,F1) and (E,F2) have the WMP, then so does (E,F1 ⊕∞ F2).

Proof. (a) and (d) are immediate. Let us prove (b). Let T ∈ L(E/E1, F ) be givenand let (xn)n∈N be a non-weakly null maximizing sequence in SE/E1 for T . Consider

Q ∈ L(E,E/E1) to be the quotient map and define T := T ◦ Q ∈ L(E,F ). Then,

‖Q‖ = 1 and ‖T‖ = ‖T‖. Now, since 1 = ‖xn‖ = inf{‖x‖ : x ∈ xn}, for each n ∈ N,we have that there exists xn ∈ xn such that 1 6 ‖xn‖ 6 1 + 1/n. This implies that the

sequence (yn)n∈N := (xn/‖xn‖)n∈N ⊂ SE is a non-weakly null maximizing sequence for T

4 DANTAS, JUNG, AND MARTINEZ-CERVANTES

since

‖T (yn)‖ =1

‖xn‖‖T (xn)‖ =

1

‖xn‖‖T (Q(xn))‖ =

1

‖xn‖‖T (xn)‖ −→ ‖T‖ = ‖T‖.

Thus, T attains its norm at some x ∈ SE. This implies that ‖T‖ = ‖T‖ = ‖T (x)‖ =‖T (Q(x))‖ and since Q(x) ∈ BE/E1 , T attains its norm at Q(x).

For item (c), let us suppose that T ∈ L(E1 ⊕1 E2, F ) is a norm one operator for whichthere exists a non-weakly null maximizing sequence (xn, yn)n∈N ⊂ SE1⊕1E2 but T does notattain its norm. Without loss of generality, we suppose that (xn)n∈N ⊂ BE1 is non-weaklynull and ‖xn‖ > ε for every n ∈ N, where ε > 0. Notice that

‖T (x, y)‖ = ‖T |E1(x) + T |E2(y)‖ 6 ‖T |E1‖‖x‖+ ‖T |E2‖‖y‖for every (x, y) ∈ E1⊕1E2. Bearing in mind that, on the sphere, ‖x‖+‖y‖ = 1, it followsfrom the previous inequality that 1 = ‖T‖ = ‖T |E1‖ or 1 = ‖T‖ = ‖T |E2‖. Moreover,since ‖xn‖ > ε for every n ∈ N, we have necessarily 1 = ‖T‖ = ‖T |E1‖. Thus, T |E1

does not attain its norm. Furthermore, since the sequence (xn/‖xn‖)n∈N ⊂ SE1 is non-weakly null, we conclude that it cannot be a maximizing sequence for T |E1 . Thus, thereis 0 6 C < 1 such that ‖T (xn)‖ < C‖xn‖ for n large enough. Since ‖T |E2‖ 6 ‖T‖ = 1,we conclude that

‖T (xn, yn)‖ 6 ‖T (xn)‖+ ‖Tyn‖< C‖xn‖+ ‖yn‖= ‖xn‖+ ‖yn‖ − (1− C)‖xn‖= 1− (1− C)‖xn‖6 1− (1− C)ε.

This implies that 1 = ‖T‖ 6 1− (1− C)ε, a contradiction. �

It is worth mentioning that items (c) and (d) of Proposition 2.2 hold also for finite `1-and `∞-sums. Nevertheless, this is not the case for arbitrary `1- and `∞-sums in general.Indeed, it is clear that the pair (R, F ) has the WMP for every Banach space F but (`1, F )does not. On the other hand, if the pair (E, `∞) has the WMP, then (E,F ) has the WMPfor every separable Banach space by Proposition 2.2(a). We conjecture that such a Banachspace E must be finite-dimensional. Proposition 2.6 provides some partial answers to thisconjecture.

Regarding item (b) of Proposition 2.2, it is also worth mentioning that if E1 is a closedsubspace of E, and the pairs (E1, F ) and (E/E1, F ) have the WMP, then the pair (E,F )does not necessarily have the WMP. This is so because the pair (`2⊕p R, `2) fails to haveit for p > 2 (see the proof of Corollary 2.3 below). Concerning item (a) of Proposition 2.2,we do not know whether the analogous case for domain spaces holds true (see Question4.1).

As promised, we present the following negative results concerning [2, Question 3.2] bycombining Theorem 2.1 and Proposition 2.2.

Corollary 2.3. There are reflexive Banach spaces E1, E2, and E3, all of them isomorphicto `2, such that (`2, E1), (E2, `2), and E3 fail the WMP.

SOME REMARKS ON THE WEAK MAXIMIZING PROPERTY 5

Proof. It is well-known that there exist non-norm-attaining operators in L(`2). Thus, wecan apply Theorem 2.1 to get that the pairs (`2, `2 ⊕q R) and (`2 ⊕p R, `2) fail the WMPwhenever 2 < p 6∞ and q < 2. The fact that `2 ⊕∞ R fails the WMP now follows fromProposition 2.2. �

Let us a take a brief moment to discuss when the hypothesis of Theorem 2.1 is fulfilled.That is, given two Banach spaces E and F , when can we guarantee the existence of anoperator T ∈ L(E,F ) which does not attain its norm? In order to discuss the existence ofsuch operators, recall that a Banach space E is said to have the approximation property(AP, for short) if for every compact subset K ⊂ E and every ε > 0, there exists afinite-rank operator T on E such that ‖T (x) − x‖ < ε for every x ∈ K. Also, a Banachspace E is said to have the compact approximation property (CAP, for short) if for everycompact set K ⊂ E and every ε > 0, there exists a compact operator T on E such that‖T (x)− x‖ < ε for every x ∈ K. It is clear that the AP implies the CAP while G. Willisproved in [16] that there exists a Banach space having the CAP but failing the AP.

J.R. Holub showed in [7] that if E and F are both reflexive spaces such that E or Fhas the AP, then the condition L(E,F ) = NA(E,F ) is equivalent to L(E,F ) = K(E,F ),which in turn is equivalent to the fact that the Banach space L(E,F ) is reflexive byusing, elegantly we must say, the James Theorem. Moreover, let us point out that thisequivalence is generalized by J. Mujica in [13, Theorem 2.1] to the case when E or F hasthe CAP instead of the AP. We will be using these facts in the next results. For furtherdiscussions on Holub and Mujica’s results, we refer the reader to the recent paper [5],where these results are generalized and, moreover, some conditions are given so that theexistence of operators that do not attain their norms is ensured.

Back to some consequences of Theorem 2.1, we see that there exist operators fromLp[0, 1] into Lq[0, 1] for 1 < p, q < ∞ which do not attain their norms by combin-ing [15, Theorem A2] with [7, Theorem 2]. Thus, we can deal with pairs of the form(Lp[0, 1], Lq[0, 1]). Nevertheless, we do not know whether the pair (Lp[0, 1], Lq[0, 1]) hasthe WMP when 1 < p 6 2 6 q <∞ (except for p = q = 2) (see Question 4.2).

Theorem 2.4. If p > 2 or q < 2, then (Lp[0, 1], Lq[0, 1]) does not satisfy the WMP.

Proof. We suppose first that p > 2. Let us notice that `2 embeds isometrically into Lq[0, 1](see, e.g., [1, Proposition 6.4.13]). Assume to the contrary that the pair (Lp[0, 1], Lq[0, 1])satisfies the WMP. Then, by Proposition 2.2, we have that (Lp[0, 1], `2) has the WMP. Onthe other hand, by using [15, Theorem A2] and [7, Theorem 2], there exists an operatorfrom Lp[0, 1] to `2 which does not attain its norm. By Theorem 2.1, we have that thepair (Lp[0, 1] ⊕p Lp[0, 1], `2 ⊕2 `2) does not have the WMP. Since Lp[0, 1] ⊕p Lp[0, 1] and`2⊕2 `2 are isometric to Lp[0, 1] and `2 respectively, we conclude that (Lp[0, 1], `2) fails tohave the WMP, a contradiction.

Suppose now that q < 2. Without loss of generality, 1 < p <∞, so Lp[0, 1] is reflexive.Let Z be a subspace of Lp[0, 1]∗ which is isometric to `2 (see [1, Theorem 6.4.13]). Then, Z∗

is isometric to Lp[0, 1]∗∗/Z⊥ = Lp[0, 1]/Z⊥. In particular, Lp[0, 1] has a quotient isometricto `2. Suppose by contradiction that the pair (Lp[0, 1], Lq[0, 1]) satisfies the WMP. ByProposition 2.2.(b), the pair (`2, Lq[0, 1]) satisfies the WMP. This in turn implies that the

6 DANTAS, JUNG, AND MARTINEZ-CERVANTES

pair (`2⊕2 R, `2⊕q R) has the WMP, since `2⊕q R is a subspace of Lq[0, 1]. This yields acontradiction due to Theorem 2.1.(a). �

Remark 2.5. Recall that if a measure µ is not atomic then the space Lp(µ) contains a1-complemented isometric copy of Lp[0, 1] (see, e.g., pages 210 and 211 in [15]). Thus, asimple combination of Theorem 2.4 and Proposition 2.2 yields that the pair (Lp(µ), Lq(ν))does not satisfy the WMP whenever the measures µ and ν are not atomic and p > 2 orq < 2.

Let us move on with more consequences of Theorem 2.1. It seems to be natural toknow whether there exists an infinite dimensional Banach space E such that the pairs(E,F ) satisfy the WMP for every Banach space F . Analogously, we could wonder whetherthere exists an infinite dimensional Banach space F such that the pairs (E,F ) satisfy theWMP for every reflexive Banach space E. Concerning the first question, the followingresult suggests that this property might imply that E is finite dimensional.

Proposition 2.6. Let E be a Banach space. Suppose that the pair (E,F ) satisfies theWMP for every Banach space F . If

(a) E satisfies the CAP and it is isometric to E ⊕p E for some p ∈ (1,∞) or(b) E∗ has the Kadec-Klee property, i.e. if weak and norm convergent sequences co-

incide on SE∗ (in particular if E has Frechet differentiable norm or LUR norm),

then E is finite dimensional.

Proof. (a). Let us suppose that E is an infinite dimensional Banach space. Then, thereexists a non-compact operator on E into itself; hence the result [13, Theorem 2.1] fromJ. Mujica asserts that there exists an operator S ∈ L(E) which does not attain its norm.It follows that the pair (E ⊕p E,E ⊕q E) fails to have the WMP for every 1 6 q < p byTheorem 2.1. Since E is isometric to E ⊕p E, it follows that the pair (E,E ⊕q E) alsofails the WMP, which is a contradiction.

(b). Let (xn)n∈N ⊂ SE be given. Since E is reflexive, we may assume that xnw−→ u for

some u ∈ BE.

Claim: We have that either u = 0 or u ∈ SE.

The claim implies that E must be finite dimensional, otherwise, since E is reflexive,the unit sphere SE would be weakly sequentially dense in BE by Josefson-Nissenzweigtheorem (see, for instance, [6, Chapter XII, pg. 219]).

Let us now prove the claim. For each n ∈ N, let x∗n ∈ SE∗ be such that x∗n(xn) = 1.

Passing to a subsequence if necessary, we may assume that x∗nw−→ x∗∞ for some x∗∞ ∈ BE∗ .

Consider the linear operator T ∈ L(E, `∞) defined by T (x) := (αnx∗n(x))n∈N for x ∈ E,

where (αn)n∈N ⊂ (0, 1) converges to 1. Let us notice that ‖T‖ = 1 and (xn)n∈N is amaximizing sequence for T . Assume that u 6= 0. By assumption, the pair (E, `∞) hasthe WMP, which implies that T attains its norm. So, there exists x∞ ∈ SE such that‖T (x∞)‖ = 1. This means that supn∈N αn|x∗n(x∞)| = 1 = |x∗∞(x∞)|. Thus, ‖x∗∞‖ = 1and, by the Kadec-Klee property, (x∗n)n∈N is norm convergent to x∗∞. Therefore, it follows

SOME REMARKS ON THE WEAK MAXIMIZING PROPERTY 7

that

|x∗∞(u)| = limj|x∗∞(xn)| = lim

j|x∗n(xn)| = 1,

which implies that ‖u‖ = 1 and this proves the claim and, consequently, item (b). �

It is immediate from Proposition 2.6(a) that if H is Hilbert space and (H,F ) has theWMP for every Banach space F , then H must be finite dimensional. We do not knowwhether the assumption of (E,F ) having the WMP for every Banach space F impliesthat E is finite dimensional in general (see Question 4.3).

For range spaces, we have the following characterization.

Theorem 2.7. Let F be a Banach space. Then, the following statements are equivalent.

(a) the pair (E,F ) satisfies the WMP for every reflexive space E;(b) F has the Schur property.

Proof. If F has the Schur property, then every operator from a reflexive Banach spaceE to F is compact; hence the pair (E,F ) has the WMP for every reflexive space E.Conversely, assume that F does not satisfy the Schur property. By [5, Theorem 3.10],there exist a reflexive space E and an operator T : E −→ F which does not attain itsnorm. Now, Theorem 2.1 shows that the pair (E ⊕∞ R, F ) fails to have the WMP, acontradiction. �

It is worth mentioning that every pair of the form (`p, c0) satisfies the WMP for 1 < p <∞ (see [4, Corollary 3.9]). However, as the space c0 does not have the Schur property, itfollows from Theorem 2.7 that there exists an infinite dimensional reflexive space E suchthat (E, c0) fails to have the WMP. Moreover, it is known by [11, Lemma 2.2] (see also [3,Proposition 4.3]) that if E is an infinite dimensional space, then there exists an operatorT from E into c0 which does not attain its (quasi-) norm (see the comments precedingDefinition 2.15 for the definition of quasi norm-attaining operators). On the other hand,we shall see that for any reflexive space E, if T ∈ L(E, c0) and K ∈ K(E, c0) satisfies that‖T‖ < ‖T +K‖, then T +K attains its norm (see Question 4.4).

Proposition 2.8. Let E be an infinite dimensional reflexive space. If T ∈ L(E, c0) andK ∈ K(E, c0) satisfy that ‖T‖ < ‖T +K‖, then T +K attains its norm. However, thereexists an isomorphic copy E ′ of E such that the pair (E ′, c0) fails to have the WMP.

Proof. Notice that if T ∈ L(E, c0), then T (x) = ((T ∗(e∗n))(x))n∈N for every x ∈ E, where(e∗n)n∈N ⊂ `1 is the unit vector basis and (T ∗(e∗n))n∈N is a weakly-null sequence. Moreover,as K ∈ L(E, c0) is compact (hence, so is its adjoint K∗), we have that (K∗(e∗n))n∈Nconverges to 0 in norm. Suppose that ‖T‖ < ‖T + K‖. From the facts that ‖T + K‖ =supn∈N ‖(T ∗ + K∗)(e∗n)‖ and that (K∗(e∗n))n∈N converges in norm to zero it follows theexistence of n0 ∈ N such that ‖T +K‖ = ‖(T ∗+K∗)(e∗n0

)‖. By the reflexivity of E, thereis x0 ∈ SE such that ‖(T ∗+K∗)(e∗n0

)‖ = |((T ∗+K∗)(e∗n0))(x0)|, which implies that T +K

attains its norm at x0. For the second part, as K is 1-complemented in E, we have thatE is isomorphic to E ′ := E1 ⊕∞ K for some infinite dimensional Banach space E1. LetT : E1 −→ c0 be a norm-one operator which does not attain its norm (see [11, Lemma2.2]). By Theorem 2.1, we get that the pair (E ′, F ) fails to have the WMP. �

8 DANTAS, JUNG, AND MARTINEZ-CERVANTES

From Proposition 2.8, we have the following consequence.

Corollary 2.9. Let E be a Banach space. If (E ′, F ) has the WMP for every isomorphiccopy E ′ of E and for every Banach space F , then E must be finite dimensional.

Motivated from the fact [14, Theorem 1] that the pair (`p, `q) has the WMP for every1 < p, q <∞, we wonder whether the pairs of the form (`s⊕p `p, `s⊕q `q) have the WMPor not for 1 < p, q, s < ∞. As a matter of fact, as a direct consequence of Theorem 2.1,we have that the pair (`s ⊕p `p, `s ⊕q `q) fails the WMP whenever p > q and 1 < s <∞.Moreover, the pair (`s⊕p `p, `s⊕q `q) does not have the WMP whenever s < p 6 q. Indeed,if it has the property, then so does (`s ⊕p `p, `s). This implies that (`s ⊕p R, `s ⊕s R) hasthe WMP, which contradicts Theorem 2.1. Similarly, one can show that if 1 < p 6 q < s,then the pair (`s ⊕p `p, `s ⊕q `q) does not have the WMP. Consequently, we have thefollowing result.

Corollary 2.10. The pair (`s ⊕p `p, `s ⊕q `q) fails the WMP whenever

(1) 1 < q < p <∞ and 1 < s <∞;(2) 1 < s < p 6 q <∞;(3) 1 < p 6 q < s <∞.

Let us notice that the case 1 < p 6 s 6 q < ∞ is not covered by Corollary 2.10.Indeed, we shall see that the pair (`s ⊕p `p, `s ⊕q `q) does have the WMP for 1 < p 6 s 6q < ∞ (considering those spaces as real Banach spaces) and we can provide a completecharacterization for the pairs (`s ⊕p `p, `s ⊕q `q) by calculating the moduli of asymptoticuniform convexity and smoothness when 1 < p 6 s 6 q < ∞. We have the followingtheorem.

Theorem 2.11. The pair of real Banach spaces (`s ⊕p `p, `s ⊕q `q) satisfies the WMP ifand only if 1 < p 6 s 6 q <∞.

In order to prove Theorem 2.11, let us present the necessary definitions we need forsuch. For a real Banach space E, let us consider the following moduli. Let t > 0 andx ∈ SE be fixed.

δE(t, x) := supdim(E/F )<∞

infy∈SF

‖x+ ty‖ − 1 and ρE(t, x) := infdim(E/F )<∞

supy∈SF

‖x+ ty‖ − 1,

where F is a finite codimensional subspace of E. Then the modulus of asymptotic uniformconvexity of E is given by

δE(t) = infx∈SE

δE(t, x)

and the modulus of asymptotic uniform smoothness of E is given by

ρE(t) = supx∈SE

ρE(t, x).

We refer the reader to [8, 12] for a background of the moduli of asymptotic uniformconvexity and smoothness. The following result seems to be new in the literature.

Theorem 2.12. Let 1 < p 6 s 6 q <∞. Then,

δ`s⊕p`p(t) = ρ`s⊕q`q(t) = (1 + ts)1s − 1

for every t > 0. In particular, ρ`s⊕q`q(t) > t− 1 for every t > 1.

SOME REMARKS ON THE WEAK MAXIMIZING PROPERTY 9

Once this result is established, Theorem 2.11 is proved automatically as a consequenceof [4, Theorem 3.1]. In order to prove Theorem 2.12, we need the following two straight-forward lemmas.

Lemma 2.13. Let r > 1, s > 0, A > 0 and B > 0 be given. Let ε > 0. Consider thecontinuous function f : [0, 1] −→ R defined by

f(t) = [(A+ εst)rs + εr(1− t

rs ) +B]

1r .

Therefore,

(i) if 1 < r < s, then f attains its minimum at t = 1.(ii) if r > s, then f attains its maximum at t = 1.

(iii) if r = s, then f ≡ (A+B + εs)1s .

Proof. For every t ∈ [0, 1], let us set

h(t) := εs(A+ εst)rs−1 − εrt

rs−1.

Then,

f ′(t) = h(t) · 1

s

[(A+ εst)

rs + εr(1− t

rs ) +B

] 1r−1.

Notice that

εr−s · trs−1 = εr−s · t

r−ss = (εs · t)

r−ss ;

hence

h(t) = εs[(A+ εst)

r−ss − (εst)

r−ss

].

From this, we can conclude items (i) and (ii). Item (iii) is immediate. �

Similar calculations yield the following result.

Lemma 2.14. Let r > 1, s > 0, and ε > 0. Consider the continuous function g :[0, 1] −→ R defined by

g(t) := [(t+ εs)rs + (1− t

rs )]

1r .

Therefore,

(i) if 1 < r < s, then g attains its minimum at t = 1.(ii) if r > s, then g attains its maximum at t = 1.

(iii) if r = s, then g ≡ (1 + εs)1s .

Now we are ready to prove Theorem 2.12.

Proof of Theorem 2.12. Let 1 < p 6 s 6 q <∞. Let us show that δ`s⊕p`p(t) > (1+ ts)1s −

1. Let t > 0 be given. Fix n ∈ N and consider the element

x0 := ((α1, . . . , αn, 0, 0, . . .), (β1, . . . , βn, 0, 0, . . .)) ∈ S`s⊕p`p .

This implies that

1 = ‖x0‖`s⊕p`p = (|α1|s + . . .+ |αn|s)ps + |β1|p + . . .+ |βn|p.

10 DANTAS, JUNG, AND MARTINEZ-CERVANTES

Set A := |α1|s + . . .+ |αn|s and B := |β1|p + . . .+ |βn|p. Then, we have that Aps +B = 1,

i.e., B = 1 − A ps . Consider the following finite codimensional subspace of `s ⊕p `p given

by

Y0 :=

z ∈ `s ⊕p `p : z = ((0, . . . , 0︸ ︷︷ ︸n-times

, a1, a2, . . .), (0, . . . , 0︸ ︷︷ ︸n-times

, b1, b2, . . .)) for ai, bi ∈ R, i ∈ N

.

Then, Y0 is a subspace of codimension 2n. Let

z = ((0, . . . , 0, a1, a2, . . .), (0, . . . , 0, b1, b2, . . .)) ∈ SY0be given. Now notice that

1 = ‖(0, . . . , 0, a1, a2, . . .)‖ps + ‖(0, . . . , 0, b1, b2, . . .)‖pp =

(∞∑j=1

|aj|s) p

s

+∞∑j=1

|bj|p

and that

‖x0 + tz‖`s⊕p`p =(‖(α1, . . . αn, ta1, ta2, . . .)‖ps + ‖(β1, . . . , βn, tb1, tb2, . . .)‖pp

) 1p

=

( n∑i=1

|α|si + ts∞∑j=1

|aj|s) p

s

+

(n∑i=1

|βi|p + tp∞∑j=1

|bj|p) 1

p

=

(A+ ts∞∑j=1

|aj|s) p

s

+B + tp

1−

(∞∑j=1

|aj|s) p

s

1p

Now, by Lemma 2.13, we have that

infz∈SY0

‖x0 + tz‖`s⊕p`p = inf06ξ61

[(A+ tsξ)

ps +B + tp(1− ξ

ps )] 1

p

= [(A+ ts)ps +B]

1p

= [(A+ ts)ps + (1− A

ps )]

1p

and applying Lemma 2.14, we get that

infz∈SY0

‖x0 + tz‖`s⊕p`p > (1 + ts)1s

Therefore, for every x ∈ S`s⊕p`p with finite support, we have that

δ`s⊕p`p(t, x) = supdim(X/Y )<∞

infz∈SY

‖x+ tz‖`s⊕p`p − 1 > (1 + ts)1s − 1.

Since δ`s⊕p`p(t, x) = limn→∞ δ`s⊕p`p(t, Pn(x)), where Pn is the projection onto the span

of the first n coordinates (see [12, Theorem 3.1, pg. 112]), we have that δ`s⊕p`p(t, x) >

(1 + ts)1s − 1 for every x ∈ S`s⊕p`p . Therefore,

δ`s⊕p`p(t) > (1 + ts)1s − 1.

SOME REMARKS ON THE WEAK MAXIMIZING PROPERTY 11

On the other hand, mutatis mutandis, we can prove that ρ`s⊕q`q(t) 6 (1+ts)1s −1. Finally,

since `s⊕p`p and `s⊕q `q contain an isometric copy of `s, we have that δ`s⊕p`p(t) 6 δ`s(t) =

(1 − ts) 1s − 1 and ρ`s⊕q`q(t) > ρ`s(t) = (1 + ts)

1s − 1 for every t > 0. These give us the

desired equalities. �

Let us finish this section by discussing some variants of the WMP. In [4], the authorsconsidered the bidual-WMP, that is, a pair (E,F ) of Banach spaces has the bidual-WMPif for every operator T ∈ L(E,F ), the existence of a non-weakly null maximizing sequencefor T implies that T ∗∗ ∈ NA(E∗∗, F ∗∗) (see [4, Definition 5.13]). This seems to be naturalto be considered due to Lindenstrauss result [10, Theorem 1], where he proved that theset of all operators in which their second adjoints are norm-attaining is dense in L(E,F ).

We introduce a property which is formally in the middle between WMP and bidual-WMP. Recall that an operator T ∈ L(E,F ) quasi attains its norm, recently defined in[3], if there exists a sequence (xn)n∈N in SE such that T (xn) −→ u in norm for someu ∈ F with ‖u‖ = ‖T‖. We denote by QNA(E,F ) the set of all quasi norm-attainingoperators from E into F . In general, we have that NA(E,F ) ⊂ QNA(E,F ) ⊂ L(E,F ).Furthermore, we have that K(E,F ) ⊂ QNA(E,F ).

Definition 2.15. We say that the pair (E,F ) has the quasi weak maximizing property(quasi-WMP, for short) if for every T ∈ L(E,F ), the existence of a non-weakly nullmaximizing sequence for T implies that T ∈ QNA(E,F ).

The following result shows some relation between the WMP, the bidual-WMP and thequasi-WMP.

Proposition 2.16. Let E and F be Banach spaces.

(a) If the pair (E,F ) has the quasi-WMP, then it also has the bidual-WMP.(b) If F is reflexive, then (E,F ) has the quasi-WMP if and only if it has the bidual-

WMP.

Moreover, if E is reflexive, then (E,F ) has the

(c) WMP if and only if it has the quasi-WMP if and only if it has the bidual-WMP.

Proof. Notice that if T ∈ QNA(E,F ), then T ∗ ∈ NA(F ∗, E∗) (see [3, Proposition 3.3])and then T ∗∗ ∈ NA(E∗∗, F ∗∗). This proves (a). On the other hand, if T ∈ L(E,F )is weakly compact and T ∗∗ attains its norm, then T ∈ QNA(E,F ) (see [3, Proposition5.1]); hence (b) is proved. Next, if E is reflexive, then NA(E,F ) = QNA(E,F ) (see [3,Proposition 4.1]) and if T attains its norm, then so does T ∗∗. �

It is natural to ask whether the bidual-WMP or the quasi-WMP of a pair (E,F )implies the reflexivity of E. In general, if L(E,F ) = K(E,F ) then (E,F ) has the quasi-WMP (and therefore the bidual-WMP), and this is the case for some pairs with E beingnonreflexive, such as (E,F ) with E nonreflexive and F finite dimensional or (C(K), `p)whenever 1 6 p < 2 (see, for example, [1] or [15]).

12 DANTAS, JUNG, AND MARTINEZ-CERVANTES

3. Proof of Theorem 2.1

We dedicate this section for the proof of Theorem 2.1. To do so, we need first thefollowing straightforward lemma.

Lemma 3.1. Let 1 6 q < p <∞. The function f : [0, 1] −→ R defined by

f(a) :=q

√(1− ap)

qp + aq (a ∈ [0, 1])

attains its absolute maximum at some m ∈ (0, 1) with f(m) > 1.

Proof. Let us notice that f is continuous on the compact interval [0, 1] and f(0) = f(1) =1. Moreover, since p > q, we have that 1/q − 1/p > 0 and then

f

(1

21p

)=

q

√(1− 1

2

) qp

+

(1

21p

)q=

q

√2

(1

2

) qp

= 21q− 1

p > 1.

Therefore, the absolute maximum is attained at some m ∈ (0, 1) and f(m) > 1. �

Now, we are ready to prove Theorem 2.1.

Proof of Theorem 2.1. It is enough to prove that the pair (E ⊕p K, F ⊕q K) does notsatisfy the WMP by using Proposition 2.2. For simplicity, we assume that K = R (thecase when K = C is proved similarly). Let us first assume that 1 6 q < p < ∞ and weprove the case p =∞ and 1 6 q <∞ afterwards.

Let 1 6 q < p <∞. Suppose that there is T 6∈ NA(E,F ) with ‖T‖ = 1. Let us definethe bounded linear operator S : E ⊕p R −→ F ⊕q R given by

S(x, a) := (T (x), a) ((x, a) ∈ E ⊕p R).

Consider the function f defined in Lemma 3.1 and m ∈ (0, 1) where f attains its absolutemaximum with f(m) > 1. We finish the first part of this proof once we prove the followingthree claims.

Claim 1: ‖S‖ 6 f(m).

Indeed, since ‖T‖ = 1, we have that

‖S‖ = sup{‖S(x, a)‖q : ‖(x, a)‖p = 1} = sup{‖(T (x), a)‖q : ‖(x, a)‖p = 1}= sup{ q

√‖T (x)‖q + |a|q : ‖x‖p + |a|p = 1}.

6 sup{ q√‖T‖q‖x‖q + |a|q : ‖x‖p = 1− |a|p}

= sup{ q

√(1− |a|p)

qp + |a|q : 0 6 |a| 6 1}

= f(m).

Claim 2: For every maximizing sequence (xn)n∈N for T , we have that ((mxn,p√

1−mp))n∈Nis a non-weakly null maximizing sequence for S.

Indeed, let us first notice that the sequence ((mxn,p√

1−mp))n∈N is non-weakly nullsince p

√1−mp 6= 0. Let us notice also that (mxn,

p√

1−mp) ∈ SE⊕pR for every n ∈ N

SOME REMARKS ON THE WEAK MAXIMIZING PROPERTY 13

since

‖(mxn, p√

1−mp)‖pp = ‖mxn‖p + 1−mp = mp + 1−mp = 1.

Finally, since

‖S(mxn,p√

1−mp)‖q = ‖(T (mxn), p√

1−mp)‖q =q

√mq‖T (xn)‖q + (1−mp)

qp

and ‖T (xn)‖ −→ ‖T‖ = 1 as n→∞, we have that

‖S(mxn,p√

1−mp)‖qn→∞−→ q

√mq‖T‖q + (1−mp)

qp =

q

√mq + (1−mp)

qp = f(m).

This shows that ‖S‖ > f(m) and by Claim 1, ‖S‖ = f(m). Moreover, ((mxn,p√

1−mp))n∈Nis a maximizing sequence for S, which is non-weakly null.

Claim 3: S does not attain the norm.

Let us notice first that any norm-one element in E⊕pR can be written as (λx,± p√

1− λp)with x ∈ SE and 0 6 λ 6 1. By contradiction, suppose that S is norm-attaining. Thus,there exist x ∈ SE and 0 6 λ 6 1 such that ‖S(λx,± p

√1− λp)‖q = ‖S‖ = f(m) (notice

that, in this case, we must have necessarily λ 6= 0 since ‖S‖ = f(m) > 1). So, we have

‖S(λx,± p√

1− λp)‖q = ‖(T (λx),± p√

1− λp)‖q

=q

√λq‖T (x)‖q + (1− λp)

qp

= f(m).

Nevertheless, since ‖T (x)‖ < ‖T‖ = 1, we have that

q

√λq‖T (x)‖q + (1− λp)

qp <

q

√λq + (1− λp)

qp 6 f(m),

which is a contradiction.

Now, let us consider the case p = ∞. Again, by Proposition 2.2, it is enough toprove that the pair (E ⊕∞ R, F ) fails the WMP. By assumption, we can pick an operator

T ∈ L(E,F ) which does not attain its norm. Define T : E⊕∞R −→ F by T (x, a) := T (x)

for every (x, a) ∈ E⊕∞R. Then, ‖T‖ = ‖T‖ and T does not attain its norm. Nevertheless,if (xn)n∈N ⊂ SE is a maximizing sequence for T , then (xn, 1)n∈N ⊂ SE⊕∞R is a maximizing

sequence for T which is non-weakly null. So, the pair (E ⊕∞ R, F ) fails the WMP. �

4. Open Questions

In this last section, we present and discuss shortly a list of problems on the weakmaximizing property.

In Proposition 2.2, we prove that if the pair of Banach spaces (E,F ) has the WMPand F1 is any subspace of F , then so does (E,F1). It seems that every known pair (E,F )with the WMP satisfies that (E1, F ) also has the WMP for every subspace E1 6 E. Wewonder whether this is always the case.

Question 4.1. Let E, F be Banach spaces and E1 be a closed subspace of E. If (E,F )has the WMP, does it also (E1, F )?

14 DANTAS, JUNG, AND MARTINEZ-CERVANTES

It follows from Theorem 2.4 that if there is a pair of the form (Lp[0, 1], Lq[0, 1]) with theWMP (different from the pair (L2[0, 1], L2[0, 1])), then p and q must satisfy the inequalities1 < p 6 2 6 q < ∞. As a matter of fact, in [4], the authors prove that there existrenormings |·|p and |·|q on Lp[0, 1] and Lq[0, 1], respectively, such that the pair ((Lp[0, 1], |·|p), (Lq[0, 1], | · |q)) satisfies the WMP for 1 < p 6 2 6 q < ∞. Nevertheless, we do notknow what happens for the standard p- and q-norms in this case. We highlight thisquestion.

Question 4.2. Suppose that 1 < p 6 2 6 q < ∞. Does the pair of Banach spaces(Lp[0, 1], ‖ · ‖p, Lq[0, 1], ‖ · ‖q) have the WMP?

In Proposition 2.6, we prove that, in several situations, the assumption that the pair(E,F ) has the WMP for every Banach space F implies that E must be finite dimensional.We wonder if this is always the case.

Question 4.3. Let E be a reflexive Banach space. If (E,F ) has the WMP for everyBanach space F , then is E finite dimensional?

In [2, Proposition 2.4], it was shown that if a pair (E,F ) of Banach spaces has the WMP,K ∈ K(E,F ), T ∈ L(E,F ), and ‖T‖ < ‖T+K‖, then T+K is norm-attaining. Moreover,we observed in Proposition 2.8 that every pair (E, c0) satisfies that if K ∈ K(E, c0),T ∈ L(E, c0), and ‖T‖ < ‖T + K‖, then T + K is norm-attaining. However, we do notknow whether the following is true or not.

Question 4.4 (R. Aron). Suppose that E and F are reflexive Banach spaces with thefollowing property: whenever T ∈ L(E,F ) and K ∈ K(E,F ) are such that ‖T‖ < ‖T +K‖, we have that T + K attains its norm. Does it then follow that the pair (E,F ) hasthe WMP?

Acknowledgements: The authors are grateful to Richard Aron, Luis Carlos GarcıaLirola, Jose David Rodrıguez Abellan, Miguel Martın, and Abraham Rueda Zoca forfruitful conversations on the topic of the paper.

References

[1] F. Albiac and N. Kalton, Topics in Banach Space Theory (second edition), Graduate Texts inMathematics 233, Springer, New York, 2016.

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[3] G. Choi, Y.S. Choi, M. Jung, and M. Martın, On quasi norm attaining operators betweenBanach spaces, preprint available on arXiv, https://arxiv.org/abs/2004.11025

[4] L.C. Garcıa-Lirola and C. Petitjean, On the weak maximizing property, preprint available onarXiv, https://arxiv.org/abs/2007.01939

[5] S. Dantas, M. Jung and G. Martınez-Cervantes, On the existence of non-norm-attainingoperators, preprint available on arXiv, https://arxiv.org/abs/2102.06452

[6] J. Diestel, Sequences and Series in Banach spaces, Springer, Edition 1, 1984.[7] J.R. Holub, Reflexivity of L(E,F ). Proc. Amer. Math. Soc. 39 (1973), no. 1, 175–177.[8] W.B. Johnson, J. Lindenstrauss, D. Preiss, and G. Schechtman, Almost Frechet differ-

entiability of Lipschitz mappings between infinite-dimensional Banach spaces, Proc. London Math.Soc. (3), 84 (2002), 711–746.

SOME REMARKS ON THE WEAK MAXIMIZING PROPERTY 15

[9] J. Kover, Compact perturbations and norm attaining operators, Quaest. Math. 28 (2005), no. 4,401–408.

[10] J. Lindenstrauss, On operators which attain their norm. Isr. J. Math. 1 (1963), 139–148.[11] M. Martın, J. Merı, and R. Paya, On the intrinsic and the spatial numerical range, J. Math.

Anal. Appl. 318 (2006), 175-189.[12] V.D. Milman, Geometric theory of Banach spaces. Part II. Geometry of the unit sphere. Uspekhi

Mat. Nauk, 1971, Volume 26, Issue 6(162), Pages 73–149. English translation: Russian Math. Surveys26 (1971), no. 6, 79–163.

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[15] H. P. Rosenthal, On quasi-complemented subspaces of Banach spaces, with an appendix oncompactness of operators from Lp(µ) to Lr(ν), J. Funct. Anal. 4 (1969), 176–214.

[16] G. Willis, The compact approximation property does not imply the approximation property. Stud.Math. 103, (1992), 99–108

(Dantas) Departament de Matematiques and Institut Universitari de Matematiques iAplicacions de Castello (IMAC), Universitat Jaume I, Campus del Riu Sec. s/n, 12071Castello, Spain.ORCID: 0000-0001-8117-3760

Email address: dantas@uji.es

(Jung) Department of Mathematics, POSTECH, Pohang 790-784, Republic of KoreaORCID: 0000-0003-2240-2855

Email address: jmingoo@postech.ac.kr

(Martınez-Cervantes) Universidad de Murcia, Departamento de Matematicas, Campus deEspinardo 30100 Murcia, SpainORCID: 0000-0002-5927-5215

Email address: gonzalo.martinez2@um.es