Reflexivity and non-weakly null maximizing sequences

TL;DR

This paper introduces a new reflexivity-related property crucial for understanding norm attaining operators, with applications to compact perturbations and characterizations of certain polynomials on Hilbert spaces.

Contribution

It presents a novel property linked to reflexivity that advances the characterization of norm attaining operators and applies it to operator perturbations and polynomial analysis.

Findings

New reflexivity property related to norm attaining operators

Characterization of norm attaining scalar-valued 2-homogeneous polynomials on ℓ₂

Application to compact perturbations of linear operators

Abstract

We introduce and explore a new property related to reflexivity that plays an important role in the characterization of norm attaining operators. We also present an application to the theory of compact perturbations of linear operators and characterize norm attaining scalar-valued continuous $2$-homogeneous polynomials on $\ell_{2}$.

Equations

$$\Vert T(x)\Vert =\Vert T\Vert .$$

$$P(x)=T_P(x,...,x)$$

$$\big{|}|x|^{r}-|x-1|^{r}-1\big{|}\leq C_{\varepsilon}|x-1|^{r-1}+\varepsilon^{r}+\sup_{|t-1|\leq\varepsilon}|\ |t|^{r}-1|.$$

$$\begin{split}\sum_{i}{\big{|}}|T(u^{n})(e_{i})|^{q}-|T(u^{n})(e_{i})-T(u)(e_{i})|^{q}-|T(u)(e_{i})|^{q}{\big{|}}\leq\\ C_{\varepsilon}\Delta_{n}+\delta(\varepsilon)\|T(u)\|_{\ell_{q}}^{q},\end{split}$$

$$\Vert T(u^n){\Vert }_{{\ell }_q}^q=\Vert T(u){\Vert }_{{\ell }_q}^q+\Vert T(u^n-u){\Vert }_{{\ell }_q}^q+\mathrm{o}(1).$$

$$\Vert u^n-u{\Vert }_{{\ell }_p}^p=1-\Vert u{\Vert }_{{\ell }_p}^p+\mathrm{o}(1).$$

$$\|T(u^{n})\|^{p}_{\ell_{q}}\leq\|T(u)\|^{p}_{\ell_{q}}+\|T\|^{p}\big{(}1-\|u\|^{p}_{\ell_{p}}+\mathrm{o}(1)\big{)}+\mathrm{o}(1).$$

$$1-{\Vert u^n-u\Vert }^2={\Vert u\Vert }^2+\mathrm{o}(1)$$

$${\Vert T(u^n)\Vert }^2-{\Vert T(u^n)-T(u)\Vert }^2={\Vert T(u)\Vert }^2+\mathrm{o}(1).$$

$$\Vert T\Vert <\Vert T+K\Vert ,$$

$$\underset{n⟶\infty }{\lim }\Vert T\left(x^n\right)+K\left(x^n\right)\Vert =\Vert T+K\Vert ,$$

$$\underset{n⟶\infty }{\lim }\Vert T\left(x^n\right)+K\left(x^n\right)\Vert \le \underset{n}{\sup }\Vert T(x^n)\Vert \le \Vert T\Vert <\Vert T+K\Vert ,$$

$$T:{\ell }_2\to {\ell }_2,T(x)=\left(x_1,\frac{x_2}{2},...,\frac{(n-1)x_n}{n},...\right)$$

$$K:{\ell }_2\to {\ell }_2,K(x)=\left(-x_1,0,0,...\right).$$

$$P_m(x)=x_1^m+m{x}_1^{m-2}\sum _{j=2}^{\infty }\frac{j}{j+1}{x}_j^2$$

$${\left(\sqrt{\frac{m-2}{m-1}}{e}_1+\sqrt{1-\frac{m-2}{m-1}}{e}_n\right)}_{n=2}^{\infty },$$

$$P(x)=x_1\left(\frac{x_2}{2},\frac{2x_3}{3},\frac{3x_4}{4},...\right).$$

$${\left(\frac{1}{\sqrt{2}}{e}_1+\frac{1}{\sqrt{2}}{e}_n\right)}_{n=2}^{\infty }$$

$$u_P(x)(y)=T_P(x,y).$$

$$\Vert u_P\Vert =\Vert T_P\Vert$$

$$\Vert P\Vert =\Vert T_P\Vert .$$

$$\Vert P\Vert =\Vert T_P\Vert =\Vert u_P\Vert \ge \Vert u_P(x^n)\Vert \ge \Vert T_P(x^n,x^n)\Vert =\Vert P(x^n)\Vert$$

$$\underset{n\to \infty }{\lim }\Vert u_P(x^n)\Vert =\Vert u_P\Vert .$$

$$\Vert u_P(x^0)\Vert =\Vert u_P\Vert =\Vert T_P\Vert .$$

$$|T_P(x^0,y^0)|=|u_P(x^0)(y^0)|=\Vert u_P(x^0)\Vert =\Vert T_P\Vert$$

$$\Vert P\Vert <\Vert P+K\Vert ,$$

Full text

The first and second authors were supported by MINECO and FEDER project MTM2017-83262-C2-1-P. The second author was also supported by PROMETEO/2017/102 of the Generalitat Valenciana. The third author was supported by CNPq-Grant 307327/2017-5. The fourth author was supported by FUNCAP/CNPq/PRONEX Grant 00068.01.00/15

1. Introduction

The concept of reflexivity plays an important role in functional analysis and is closely related to the theory of norm attaining operators. A classical result due to R. C. James [13, 14] asserts that a (real or complex) Banach space $E$ is reflexive if and only if all bounded linear functionals on $E$ attain their norms, i.e., the norm of a bounded functional, which is a supremum, is in fact a maximum. Since then, the relation between reflexivity and norm attainment has attracted the interest of many mathematicians (see, e.g., [2, 9, 12, 15, 16]).

Let ${E}$ and ${F}$ be Banach spaces over $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ and $\mathcal{L}(E,F)$ be the space of all bounded linear operators from $E$ into $F$. The topological dual of $E$ will be represented by $E^{\ast}$. A bounded linear operator ${T}\colon E\rightarrow F$ is said to attain its norm if there exists a norm one vector $x\in E$ so that

[TABLE]

A maximizing sequence for $T\in\mathcal{L}(E,F)$ is a sequence $(x^{n})_{n=1}^{\infty}$ in $E$ with $\|x^{n}\|=1$ for all $n$, so that $\|T(x^{n})\|$ converges to $\|T\|$ as $n\rightarrow\infty$. In this note we introduce and explore the following property:

**Weak Maximizing Property. ** A pair of Banach spaces $\left(E,F\right)$ is said to have the weak maximizing property if for any bounded linear operator $T:E\rightarrow F$, the existence of a non-weakly null maximizing sequence for $T$ implies that $T$ attains its norm. When $E=F$ we say that $E$ has the weak maximizing property.

In some sense this new property extends the notion of reflexivity, since the pair $\left(E,F\right)$ has the weak maximizing property for some $F\neq\left\{0\right\}$ if and only if $E$ is reflexive (see Corollary 2.5). For linear operators, the weak maximizing property should be explored for infinite dimensional target spaces $F$, since every maximizing sequence for a non-zero, finite rank bounded linear operator is non-weakly null. On the other hand, for homogeneous polynomials (see Section 3) the scalar case deserves attention since the analogue of the weak maximizing property does not characterize reflexive spaces.

We will show that, like reflexivity, this new concept is strongly connected to the theory of norm attaining operators.

The following result can be regarded as the first connection between the weak maximizing property and norm attaining operators.

Theorem 1.1**.**

([21, Theorem 1]) Let $1<p<\infty$, $1\leq q<\infty$ and $T\colon\ell_{p}\rightarrow\ell_{q}$ be a bounded linear operator. Then $T$ attains its norm if, and only if, there exists a non-weakly null maximizing sequence for $T$. In particular, $\left(\ell_{p},\ell_{q}\right)$ has the weak maximizing property.

This paper is organized as follows. We believe that it is important to understand the present limitations of our knowledge about the weak maximizing property. For this, we begin Section 2 with a brief outline of the proof of Theorem 1.1. We then observe that a full version of Theorem 1.1 is true for more general $\ell_{p}$ spaces. This allows us to deduce that $\left(\ell_{p}(\Gamma_{1}),\ell_{q}(\Gamma_{2})\right)$ has the weak maximizing property whenever $1<p<\infty$, $1\leq q<\infty$, where $\Gamma_{1}$ and $\Gamma_{2}$ are arbitrary index sets. In addition, an application of this result is given in the context of compact perturbations of continuous linear operators. Albeit simple, our application (see Proposition 2.4) reinforces the special role played by the weak maximizing property and substantially improves a result of J. Kover [17] on compact perturbations of operators acting on Hilbert spaces. As a consequence of James’ characterization of reflexivity and the results in this section, we conclude that if $E$ has the weak maximizing property, then $E$ is reflexive. So, it seems natural to ask whether the reflexivity of $E$ characterizes the weak maximizing property for $E$.

Any answer to this problem will lead us in interesting directions. A positive answer will show a new feature of reflexivity in the theory of norm attaining operators. On the other hand, a negative answer will highlight the importance of studying the weak maximizing property.

In the final section we investigate possible extensions of Theorem 1.1 to the setting of homogeneous polynomials. Recall that a continuous $m$-homogeneous polynomial $P\colon E\longrightarrow F$ is a mapping so that

[TABLE]

for some (unique) symmetric continuous $m$-linear mapping $T_{P}:{E\times\cdots\times E}\rightarrow F$. The space of all continuous $m$-homogeneous polynomials (with the $\sup$ norm) $P:E\rightarrow F$ is denoted by $\mathcal{P}(^{m}E;F)$. The concept of maximizing sequence is naturally extended to polynomials and an unavoidable question is whether an analogue of Theorem 1.1 is valid for polynomials. It is worth recalling that contrary to the case of linear operators, compact polynomials on reflexive Banach spaces are not necessarily norm attaining.

A pair of Banach spaces $\left(E,F\right)$ is said to have the $m-$homogeneous polynomial weak maximizing property $(m\in\mathbb{N})$ if for any continuous $m$-homogeneous polynomial $P:E\rightarrow F$, the existence of a non-weakly null maximizing sequence for $P$ implies that $P$ attains its norm.

We will show by means of a counterexample that there is no analogue of Theorem 1.1 for polynomials of degree $m\geq 3$. Surprisingly, for $m=2,$ $E=\ell_{2}$ and $F=\mathbb{K}$ a satisfactory extension of Theorem 1.1 is valid. In other words, we show in Theorem 3.1 that a $2-$homogeneous polynomial $P:\ell_{2}\to\mathbb{K}$ attains its norm if and only if there is a non-weakly null sequence $(x^{n})_{n=1}^{\infty}\subset\ell_{2}$ of unit vectors such that $|P(x^{n})|\to\|P\|$, i.e., the pair $\left(\ell_{2},\mathbb{K}\right)$ has the $2-$homogeneous polynomial weak maximizing property.

2. Weak maximizing property for pairs of $\ell_{p}(\Gamma)$ and an application

We begin this section with the promised sketch of the proof of the non-trivial direction of Theorem 1.1:

Sketch of the proof of Theorem 1.1..

Let $(u^{n})_{n=1}^{\infty}\subset\ell_{p}$ be a non-weakly null maximizing sequence for $T.$ By reflexivity, we may assume that $(u^{n})_{n=1}^{\infty}$ converges weakly to $u\neq 0.$ One can choose a further subsequence, still calling it $(u^{n})_{n=1}^{\infty},$ with the property that the sequence \big{(}|T(u^{n})(e_{i})-T(u)(e_{i})|^{q-1}\big{)}_{i=1}^{\infty} converges weakly to [math] in $\ell_{\frac{q}{q-1}}.$

Next, by calculus one shows that for fixed $r>1$ and $\varepsilon>0,$ there is a positive constant $C_{\varepsilon}$ such that the following inequality holds for every $x\in\mathbb{R}$

[TABLE]

For fixed $n,$ by applying estimate $\eqref{real}$ to $x_{i}=\frac{T(u^{n})(e_{i})}{T(u)(e_{i})}$ whenever the denominator is non-zero and adding over $i,$ we get

[TABLE]

where $\Delta_{n}=\sum_{i}|T(u)(e_{i})||T(u^{n})(e_{i})-T(u)(e_{i})|^{q-1}.$

Since the sequence $(|T(u^{n})(e_{i})-T(u)(e_{i})|^{q-1})_{i=1}^{\infty}$ is weakly null in $\ell_{\frac{q}{q-1}}$ and $T(u)\in\ell_{q}=(\ell_{\frac{q}{q-1}})^{\ast},$ it follows that $\Delta_{n}\to 0$ as $n\to\infty.$ Letting $n\to\infty$ in $\eqref{coordinates}$ and then letting $\varepsilon\to 0,$ we conclude that

[TABLE]

In a similar way, one shows that for a further subsequence $(u^{n})_{n=1}^{\infty}$ of the original sequence, one has

[TABLE]

Combining $\eqref{equality_T}$ and $\eqref{equality_u},$ we see that

[TABLE]

Let $n\to\infty,$ it easily follows that $\|T(u)\|_{\ell_{p}}\geq\|T\|\|u\|_{\ell_{p}}.$ ∎

In connection with Theorem 1.1 and its proof, the following comments are pertinent.

Remarks 2.1**.**

(a) Observe that the proof makes great use of the particular norms of both the domain and the range of the operator. In particular, the argument does not hold for renormings of either $\ell_{p}$ or $\ell_{q}$ and, indeed, the authors do not know if the result is even true for other norms.

(b) Note that the norm of the operator $T$ is attained at the normalization $\frac{u}{||u||}$ of the non-zero limit of the maximizing sequence. This should be compared with several examples in Section 3.

An important question that remains open is whether the pair $(L_{p},L_{q})$ has the weak maximizing property, for some range of exponents $(p,q)$. In the case $p\leq q$ the same argument used in the proof of Theorem 1.1 (and Proposition 2.2 below) can be adapted if one substitutes weak convergence by almost everywhere convergence of the maximizing sequence and its image.

However, for $\ell_{p}(\Gamma)$ spaces, we have a similar version to Theorem 1.1. Note that in the proof below we also prove a version of Pitt’s Theorem (see [22]) for $\ell_{p}(\Gamma)$ spaces.

Proposition 2.2**.**

If $1<p<\infty$, $1\leq q<\infty$ and $\Gamma_{1}$ and $\Gamma_{2}$ are arbitrary index sets, then the pair $\left(\ell_{p}(\Gamma_{1}),\ell_{q}(\Gamma_{2})\right)$ has the weak maximizing property.

Proof.

Let $T\colon\ell_{p}(\Gamma_{1})\rightarrow\ell_{q}(\Gamma_{2})$ be a bounded linear operator. The case $1<p\leq q<\infty$ can be obtained, mutatis mutandis, from the proof of Theorem 1.1.

A standard argument shows that when $p>q,$ any bounded linear operator attains its norm. To see this, suppose that $T\colon\ell_{p}(\Gamma_{1})\rightarrow\ell_{q}(\Gamma_{2})$ is a bounded linear operator. Repeating the standard argument, we claim that $T$ is compact. If $T$ were not compact, there would be a sequence of unit vectors $(x^{n})_{n=1}^{\infty}$ in $\ell_{p}(\Gamma_{1})$ such that the sequence $(T(x^{n}))_{n=1}^{\infty}$ has no limit point. Let $\Lambda_{n}=\{i\in\Gamma_{1}:\ x^{n,i}\neq 0\}$ and $\Lambda^{\prime}=\cup\Lambda_{n}.$ Now, consider the restriction of $T$ to $\ell_{p}(\Lambda^{\prime})$, which is essentially $\ell_{p}$ (since $\Lambda^{\prime}$ is countable). Then we would have a non-compact bounded linear operator $T\colon\ell_{p}\rightarrow\ell_{q}$. This contradicts the classical Pitt theorem and the proof is complete.

∎

Although the previous proof covers the case of Hilbert spaces, we think that the following sketch of a direct proof when $T:H_{1}\to H_{2}$, for $H_{1}$ and $H_{2}$ Hilbert spaces, deserves to be mentioned. For that, let $(u^{n})_{n=1}^{\infty}$ be a non-weakly null maximizing sequence for $T$. Passing to a subsequence if necessary, we may assume that $(u^{n})_{n=1}^{\infty}$ converges weakly to some $u\not=0.$ A simple and direct handling of inner-product properties gives us

[TABLE]

and

[TABLE]

A combination of the previous equalities completes the argument.

? Compact perturbations of continuous linear operators have important connections in functional analysis (see, e.g., [10] and the famous “Scalar-plus-compact problem” of Lindenstrauss [19], solved by Argyros and Haydon [3]). In [17], J. Kover showed the following result:

Proposition 2.3** (J. Kover [17]).**

Let $H$ be a Hilbert space and $K,T:H\rightarrow H$ compact, resp. bounded linear operators. If

[TABLE]

then $T+K$ is norm attaining.

In other words, if a compact perturbation of $T\in\mathcal{L}(H,H)$ has norm strictly greater than $T$, then this perturbation is norm attaining. We generalize this result by proving that it holds for all pairs $(E,F)$ satisfying the weak maximizing property. This result highlights the importance of the weak maximizing property and motivates the investigation of the problem stated in the Introduction. Proposition 2.2 helps to understand that Proposition 2.4 generalizes Proposition 2.3.

Proposition 2.4**.**

Suppose that the pair $\left(E,F\right)$ has the weak maximizing property and that $K,T:E\rightarrow F$ are compact, resp. bounded linear operators. If

[TABLE]

*then $T+K$ is norm attaining.

In particular, this applies when $E=\ell_{p}(\Gamma_{1})$ and $F=\ell_{q}(\Gamma_{2})$, with $1<p<\infty$ and $1\leq q<\infty$ and $\Gamma_{1}$, $\Gamma_{2}$ arbitrary sets of indexes.

Before giving the short argument, we remark that we are not really interested in $T$ or in $K$ but rather in their sum $T+K.$ The reason for this is that it may be possible that $||T||\geq||T+K||$ but, for some other bounded, resp. compact operator $T^{\prime},$ resp. $K^{\prime},$ we have $T+K=T^{\prime}+K^{\prime}$ and also that $||T^{\prime}||<||T^{\prime}+K^{\prime}||.$ In such a case, if $(E,F)$ has the weak maximizing property then $T+K=T^{\prime}+K^{\prime}$ is norm-attaining. Also, assuming that $(E,F)$ has the weak maximizing property, if $T\in\mathcal{L}(E,F)$ and $\epsilon>0$ are arbitrary, it is easy to show that there is a rank one operator $K:E\to F$ such that $\|K\|<\epsilon$ and that $\|T\|<\|T+K\|.$ Thus any $T$ can be approximated by a norm-attaining operator whose difference with $T$ has rank one. Compare with the result of Lindenstrauss in [18, Theorem 1], but of course the result itself was subsumed by Theorem 15 in Stegall’s paper [23].

Proof.

Suppose that $T+K$ fails to be norm attaining. Since every maximizing sequence $\left(x^{n}\right)_{n=1}^{\infty}$ for $T+K$ is weakly null, by the compactness of $K$ we have that $(K\left(x^{n}\right))_{n=1}^{\infty}$ converges to zero. Since $\left(x^{n}\right)_{n=1}^{\infty}$ is maximizing for $T+K$ we know that

[TABLE]

and we get

[TABLE]

which is a contradiction. ∎

If $E$ is non-reflexive, let $\varphi\in E^{\ast}$ be a functional that does not attain its norm. If $F\neq\{0\}$, choose any non-zero $y^{0}\in F.$ Then the compact operator $K:E\to F,\ K(x)=\varphi(x)y^{0}$ is not norm attaining. Taking $T=0,$ we see that Proposition 2.4 fails. This argument extends to prove the following result.

Corollary 2.5**.**

A pair $\left(E,F\right)$ has the weak maximizing property for some $F\neq\{0\}$ if and only if $E$ is reflexive. In particular if $E$ has the weak maximizing property then $E$ is reflexive.

It is also worth mentioning that the inequality in (2.6) needs to be strict. For example, consider

[TABLE]

and

[TABLE]

It is easy to check that $T+K$ is not norm attaining and $\left\|T+K\right\|=\left\|T\right\|=1.$

3. Norm attaining polynomials

The investigation of norm attaining polynomials and multilinear mappings is a subject which has attracted the attention of several authors (see [1, 4, 5, 6] and references therein).

In this section $m\geq 2$ will always denote a positive integer. If $1<p,q<\infty$ and $mq<p$, then every continuous $m$-homogeneous polynomial $P:\ell_{p}\rightarrow\ell_{q}$ is sequentially weak-to-norm continuous (see, e.g., [11, Exercise 2.65] for the more general case of multilinear mappings). Hence, since $\ell_{p}$ is reflexive, one can prove that $P$ attains its norm. So, the analogous version of Theorem 1.1 (when $mq<p$) works for $m-$homogeneous polynomials. Hence, the relevant question is whether Theorem 1.1 is true for the case $mq\geq p$. As we will show, the answer is negative in general.

3.1. The case $m\geq 3$

We remark that a counterexample for the case $F=\mathbb{R}$ also works for a general $F$ over $\mathbb{R}$. So, we will restrict ourselves to the case $F=\mathbb{R}$.

If $m\geq 3,$ the polynomial $P_{m}:\ell_{2}\rightarrow\mathbb{R}$ defined by

[TABLE]

does not attain its norm of 2\big{(}\frac{m-2}{m-1}\big{)}^{\frac{m-2}{2}}\ \ (>1=P_{m}(e_{1})), and there is a non-weakly null maximizing sequence for $P_{m}.$ In fact, one can see that $P_{m}$ does not attain its norm and the sequence

[TABLE]

where $(e_{n})_{n=1}^{\infty}$ is the canonical basis of $\ell_{2}$, is maximizing and non-weakly null.

3.2. The case $m=2$

Here we still have a counterexample when $F$ is infinite dimensional. In fact, consider $P:\ell_{2}\rightarrow\ell_{2}$ defined by

[TABLE]

Note that $\left\|P\right\|=1/2$ and $P$ does not attain its norm. On the other hand, the sequence

[TABLE]

is maximizing and non-weakly null.

3.3. The case $m=2,$ $E=\ell_{2}$ and $F=\mathbb{K}$

Now we will show that, contrary to the other two cases, the case of scalar-valued $2$-homogeneous polynomials has a positive result:

Theorem 3.1**.**

Let $P\colon\ell_{2}\rightarrow\mathbb{K}$ be a continuous $2$-homogeneous polynomial. Then $P$ attains its norm if, and only if, there exists a non-weakly null maximizing sequence for $P$. In particular, the pair $\left(\ell_{2},\mathbb{K}\right)$ has the $2-$homogeneous polynomial weak maximizing property.

Proof.

As usual, just one implication needs a proof. Let $(x^{n})_{n=1}^{\infty}$ be a non-weakly null maximizing sequence for $P.$ Let $u_{P}:\ell_{2}\rightarrow\ell_{2}^{\ast}$ be the linear operator defined by

[TABLE]

It is well-known that

[TABLE]

and since we are dealing with Hilbert spaces, an old result due to Banach (see [7] or [11, Proposition 1.44] for an easy available reference) asserts that the norm of any homogeneous polynomial coincides with the norm of its associated symmetric multilinear operator (see also [20, Theorem 2.1] for a modern constructive approach), i.e.,

[TABLE]

So we have

[TABLE]

and since $\left\|P(x^{n})\right\|$ converges to $\left\|P\right\|$ as $n$ goes to infinity, we conclude that

[TABLE]

So, by Theorem 1.1 we have that $u_{P}:\ell_{2}\rightarrow\ell_{2}^{\ast}$ attains its norm at some norm-one vector $x^{0}\in\ell_{2}$. Hence

[TABLE]

But, since $u_{P}(x^{0}):\ell_{2}\rightarrow\mathbb{K}$ is a continuous linear functional, $u_{P}(x^{0})$ also attains its norm at some norm-one vector $y^{0}\in\ell_{2}$. Therefore

[TABLE]

and so $T_{P}:\ell_{2}\times\ell_{2}\rightarrow\mathbb{K}$ attains its norm at $(x^{0},y^{0})\in\ell_{2}\times\ell_{2}$. Finally, thanks again to Banach’s result (see [7]) we conclude that $P$ attains its norm. ∎

The results in this section show that things work nicely, from the point of view of non-weakly null maximizing sequences and norm attainment, for scalar-valued $2$-homogeneous polynomials. However this is no longer the case for vector-valued $2$-homogeneous polynomials and for scalar-valued $m$-homogeneous polynomials with $m\geq 3$. This illustrates a pattern in the attempts to generalize linear results to the polynomial setting. For example, in [8, Section 4] the validity of expected results resists one degree of homogeneity more: things work nicely for $2$-homogeneous polynomials and for scalar-valued $3$-homogeneous polynomials but deteriorate for vector-valued $3$-homogeneous polynomials and for $m$-homogeneous polynomials with $m\geq 4$. Further information is found, for example, in [6].

3.4. Completely continuous perturbations of homogeneous polynomials

An $m$-homogeneous polynomial is said to be completely continuous when it is weak-to-norm continuous. For example, finite-type polynomials are always completely continuous.

¿From the proof of Proposition 2.4 one can check that a polynomial version is possible if we replace the compact perturbation by a perturbation by a completely continuous polynomial. More precisely, if the pair $(E,F)$ has the $m$-homogeneous polynomial weak maximizing property, the following holds: For any continuous $m$-homogeneous polynomial $P:E\rightarrow F$ and any completely continuous $m$-homogeneous polynomial $K:E\rightarrow F$ such that

[TABLE]

the polynomial $P+K$ is norm attaining.

Since the pair $(\ell_{2},\mathbb{K})$ has the $2$-homogeneous polynomial weak maximizing property (see Theorem 3.1), we see that completely continuous perturbations of $2$-homogeneous polynomials increasing their norms provide norm attaining polynomials. That is, if $P:\ell_{2}\rightarrow\mathbb{K}$ is a continuous $2$-homogeneous polynomial, $K:\ell_{2}\rightarrow\mathbb{K}$ is a completely continuous $2$-homogeneous polynomial and

[TABLE]

then $P+K$ is norm attaining.

We conclude with the following questions:

Questions 3.2**.**

1. (1)

Are there infinite dimensional Banach spaces $E,F_{1}$ and $F_{2}$ for which the pair $(E,F_{1})$ has the weak maximizing property but the pair $(E,F_{2})$ fails to have it? In particular, if $(E,F)$ has the weak maximizing property, does $(E,F^{\prime})$ also have it for an isomorphic copy of $F?$ More generally, if the pair $(E,F)$ has the weak maximizing property and if $E^{\prime},F^{\prime}$ are isomorphic to $E,F,$ respectively, does the pair $(E^{\prime},F^{\prime})$ have the same property? 2. (2)

What is the relation between $E$ having the weak maximizing property and $E$ being reflexive? We conjecture that the two properties are not equivalent.

Acknowledgment. We thank Geraldo Botelho for helpful conversations and María Acosta and Lawrence Harris for providing us the references [7] and [20].