Spaceability of the set of bounded linear non-absolutely summing operators in Quasi-Banach sequence spaces

TL;DR

This paper demonstrates that in certain quasi-Banach sequence spaces, there exists an infinite-dimensional subspace of bounded linear operators where every nonzero operator is non-absolutely summing, revealing a rich structure of such operators.

Contribution

It establishes the spaceability of non-absolutely summing operators in $ ext{L}( extstyle ext{ell}_p; ext{ell}_p)$ for $0<p<1$, improving previous results.

Findings

Existence of infinite-dimensional subspace of non-absolutely summing operators.

Every nonzero operator in this subspace is non $(r,s)$-absolutely summing.

The result applies for all $0<p<1$ and $1 extless s extless r< extless \infty$.

Abstract

In the short note we prove that for every $0<p<1$, there exists an infinite dimensional closed linear subspace of $\mathcal{L}\left( \ell_{p};\ell_{p}\right) $ every nonzero element of which is non $(r,s)$-absolutely summing operator for the real numbers $r,s$ with $1\leq s\leq r<\infty$. This improve a result obtained in \cite{DanielT}.

Equations

$${\Vert x\Vert }_{{\ell }_p}={\left(\sum _{k=1}^{\infty }{\left|x_k\right|}^p\right)}^{\frac{1}{p}}.$$

$${\mathbb{N}}_k=\left\{n_1^{\left(k\right)}<n_2^{\left(k\right)}<\cdots \right\}.$$

$${\ell }_p^{\left(k\right)}:=\left\{x\in {\ell }_p:x_j=0\text{ if }j\notin {\mathbb{N}}_k\right\}.$$

$$i^{\left(k\right)}:{\ell }_p⟶{\ell }_p^{\left(k\right)}$$

$$\left(i^{\left(k\right)}\left(x\right)\right)_{n_{m}^{\left(j\right)}}=\left\{\begin{array}[c]{c}x_{m},\text{ if }j=k,\\ 0,\text{ if }j\neq k\end{array}\right.$$

$${\Vert i^{\left(k\right)}\left(x\right)\Vert }_{{\ell }_p^{\left(k\right)}}={\Vert i\left(x\right)\Vert }_{{\ell }_p}$$

$$u_k:{\ell }_p\stackrel{i^{\left(k\right)}}{⟶}{\ell }_p^{\left(k\right)}\stackrel{j_k}{⟶}{\ell }_p,$$

$${\Vert i^{\left(k\right)}\left(x\right)\Vert }_{{\ell }_p^{\left(k\right)}}={\Vert i\left(x\right)\Vert }_{{\ell }_p}={\Vert u_k\left(x\right)\Vert }_{{\ell }_p}$$

$$T\left({\left(a_i\right)}_{i=1}^{\infty }\right)=\sum _{i=1}^{\infty }{a}_i{u}_i,$$

$$T\left({\ell }_p╲\left\{0\right\}\right)\subset \mathcal{L}\left({\ell }_p;{\ell }_p\right)╲\bigcup _{1\le s\le r<\infty }{\prod }_{\left(r,s\right)}\left({\ell }_p;{\ell }_p\right).$$

$$\overline{T\left({\ell }_p\right)}╲\left\{0\right\}\subset \mathcal{L}\left({\ell }_p;{\ell }_p\right)╲\bigcup _{1\le s\le r<\infty }{\prod }_{\left(r,s\right)}\left({\ell }_p;{\ell }_p\right).$$

$$\Psi =\underset{k\to \infty }{\lim }T\left({\left(a_i^{\left(k\right)}\right)}_{i=1}^{\infty }\right)\text{in}\mathcal{L}\left({\ell }_p;{\ell }_p\right).$$

$$T\left({\left(a_i^{\left(k\right)}\right)}_{i=1}^{\infty }\right)=\sum _{i=1}^{\infty }{a}_i^{\left(k\right)}{u}_i.$$

$$\Psi =\underset{k\to \infty }{\lim }\sum _{i=1}^{\infty }{a}_i^{\left(k\right)}{u}_i=\sum _{i=1}^{\infty }\underset{k\to \infty }{\lim }{a}_i^{\left(k\right)}{u}_i.$$

$$\Psi (x)=\underset{k\to \infty }{\lim }\sum _{i=1}^{\infty }{a}_i^{\left(k\right)}{u}_i(x)=\sum _{i=1}^{\infty }\underset{k\to \infty }{\lim }{a}_i^{\left(k\right)}{u}_i(x).$$

$$\underset{k\to \infty }{\lim }{a}_i^{\left(k\right)}={\alpha }_i\text{for all}i.$$

$$\sum _{j=1}^{\infty }{\left|\phi \left(x^{\left(j\right)}\right)\right|}^s<\infty \text{ and }\sum _{j=1}^{\infty }{\Vert u_i\left(x^{\left(j\right)}\right)\Vert }_{{\ell }_p}^r=\infty ,$$

$$\sum _{j=1}^{\infty }{\Vert \Psi \left(x^{\left(j\right)}\right)\Vert }_{{\ell }_p}^r$$

$$=\sum _{j=1}^{\infty }{\Vert \sum _{i=1}^{\infty }\underset{k\to \infty }{\lim }{a}_i^{\left(k\right)}{u}_i\left(x^{\left(j\right)}\right)\Vert }_{{\ell }_p}^r$$

$$=\sum _{j=1}^{\infty }{\Vert \sum _{i=1}^{\infty }{\alpha }_i.u_i\left(x^{\left(j\right)}\right)\Vert }_{{\ell }_p}^r.$$

$$\sum _{j=1}^{\infty }{\Vert \Psi \left(x^{\left(j\right)}\right)\Vert }_{{\ell }_p}^r$$

$$\ge \sum _{j=1}^{\infty }{\Vert {\alpha }_{i_0}.u_{i_0}\left(x^{\left(j\right)}\right)\Vert }_{{\ell }_p}^r$$

$$={\left|{\alpha }_{i_0}\right|}^r.\sum _{j=1}^{\infty }{\Vert u_{i_0}\left(x^{\left(j\right)}\right)\Vert }_{{\ell }_p}^r=\infty$$

$$\sum _{j=1}^{\infty }{\Vert \Psi \left(x^{\left(j\right)}\right)\Vert }_{{\ell }_p}^r=\infty .$$

$$\overline{T\left({\ell }_p\right)}╲\left\{0\right\}\subset \mathcal{L}\left({\ell }_p;{\ell }_p\right)╲\bigcup _{1\le s\le r<\infty }{\prod }_{\left(r,s\right)}\left({\ell }_p;{\ell }_p\right)$$

Full text

Daniel Tomaz is supported by Capes

1. Introduction

In the last decade many authors have been searching for large linear structures of mathematical objects enjoying certain special properties. These notions of lineability/spaceability has been investigated in several contexts, for instance, Functional Analysis, Measure Theory, Probability Theory, Set Theory, etc.

If $E$ is a vector space, a subset $A$ of $E$ is said to be lineable if $A\cup\left\{0\right\}$ contains a infinite dimensional linear subspace of $E$. Moreover, if $E$ is a topological vector space, a subset $A$ is said spaceable if $A\cup\left\{0\right\}$ contains a closed infinite dimensional linear subspace of $E$. If $\alpha$ is a cardinal number, a subset $A$ of $E$ is called $\alpha$-lineable (spaceable) if $A\cup\left\{0\right\}$ contains a (closed) $\alpha$-dimensional linear subspace of $E$.

These definitions were introduced by Aron, Gurariy and Seoane-Sepúlveda in the classical references [1] and [9], considered as the founding pillars of the theory of lineability. See also, for instance, the recent papers [3, 4, 5, 6, 7, 13]. We refer also the recent monograph [2], where many examples can be found and techniques are developed in several different frameworks.

1.1. Notation

Let us now fix some notation. Let $E,F$ be Banach or quasi-Banach spaces over the scalar field $\mathbb{K}$, which can be either $\mathbb{R}$ or $\mathbb{C}$. The space of absolutely $(r,s)$-summing linear operators from $E$ to $F$ will be represented by $\prod\nolimits_{(r,s)}\left(E;F\right)$ and the space of bounded linear operators from $E$ to $F$ will be denoted by $\mathcal{L}\left(E;F\right)$.

Recall that an linear operator $T:E\rightarrow F$ is absolutely $\left(r,s\right)$-summing if $\sum_{k}\left\|T\left(x_{k}\right)\right\|^{r}<\infty$ whenever $\left(x_{k}\right)_{k=1}^{\infty}$ is a sequence in $E$ such that $\sum_{k}\left|f\left(x_{k}\right)\right|^{s}<\infty$ for each $f\in E^{\prime}$, where $E^{\prime}$ denote the topological dual of $E$.

The basics of the linear theory of absolutely summing operators can be found in the classical book [8]. If $E$ is a Banach or quasi-Banach space, we denote by

$\ell_{p}^{w}\left(E\right)=\left\{\left(x_{j}\right)_{j=1}^{\infty}\in E^{\mathbb{N}};\text{ }{\textstyle\sum\limits_{j=1}^{\infty}}\left|\phi\left(x_{j}\right)\right|^{p}<\infty,\text{ }\forall\phi\in E^{\prime}\right\}$ the space of weakly $p$-summable $E$-valued sequences and by $\ell_{p}\left(E\right)=\left\{\left(x_{j}\right)_{j=1}^{\infty}\in E^{\mathbb{N}};\text{ }{\textstyle\sum\limits_{j=1}^{\infty}}\left\|x_{j}\right\|^{p}<\infty\right\}$ the space of absolutely $p$-summable $E$-valued sequences. We will denote by $\mathfrak{c}$ the cardinality of the continuum.

If $0<p<1$, the sequences spaces $\ell_{p}$ are quasi-Banach spaces ($p$-Banach space) with quasi-norms given by

[TABLE]

The behavior of quasi-Banach spaces or, more generally, metrizable complete topological vector spaces, called $F$-spaces is sometimes quite different from the behavior of Banach spaces. Besides, the search for closed infinite dimensional subspaces of quasi-Banach spaces is a quite delicate issue . Thus, it seems interesting to look for lineability and spaceability techniques that also cover the case of quasi-Banach spaces. For more details on quasi-Banach spaces we refer to [11]. The aim of this paper is to prove the spaceability of the set of bounded linear non-absolutely summing operators in quasi-Banach sequence spaces. To be more precise, let us to prove that $\mathcal{L}\left(\ell_{p};\ell_{p}\right)\diagdown\bigcup\limits_{1\leq s\leq r<\infty}{\textstyle\prod\nolimits_{\left(r,s\right)}}\left(\ell_{p};\ell_{p}\right)$ is $\mathfrak{c}$-spaceable for every $0<p<1$, improving a result that was proved in [14].

2. preliminaries

In this section, we will consider some common tools in the related results to the lineability/spaceability. Let us split $\mathbb{N}$ into countably many infinite pairwise disjoint subsets $\left(\mathbb{N}_{k}\right)_{k=1}^{\infty}$. For each integer $k\in\mathbb{N},$ write

[TABLE]

Define

[TABLE]

On the other hand, since $\mathbb{N=}\left\{n_{m}^{\left(j\right)}:j,m\in\mathbb{N}\right\},$ consider the sequence of linear operators

[TABLE]

given by

[TABLE]

for all $x=\left(x_{m}\right)_{m=1}^{\infty}\in\ell_{p}$. Note that

[TABLE]

where $i:\ell_{p}\longrightarrow\ell_{p}$ is the identity map. Now, for each $k\in\mathbb{N}$, consider the sequence $\left(u_{k}\right)_{k=1}^{\infty}$ in $\mathcal{L}\left(\ell_{p};\ell_{p}\right)$ defined of the form

[TABLE]

with $j_{k}:\ell_{p}^{\left(k\right)}\longrightarrow\ell_{p}$ is the inclusion operator. Moreover, notice that

[TABLE]

for all $k.$

Theorem 2.1**.**

([12, Theorem 4]) Let $0<p<1$ and $1\leq s\leq r<\infty$. Then the identity map $i:\ell_{p}\longrightarrow\ell_{p}$ is non $\left(r,s\right)$-absolutely summing.

Remark 2.2**.**

It is straightforward consequence of (2.1) and the previous theorem that for each $k\in\mathbb{N}$, the operator $u_{k}:\ell_{p}\longrightarrow\ell_{p}$ is non $\left(r,s\right)$-absolutely summing regardless of the real numbers $r,s$, with $1\leq s\leq r<\infty$.

3. The main result

Theorem 3.1**.**

$\mathcal{L}\left(\ell_{p};\ell_{p}\right)\diagdown\bigcup\limits_{1\leq s\leq r<\infty}{\textstyle\prod\nolimits_{\left(r,s\right)}}\left(\ell_{p};\ell_{p}\right)$* is $\mathfrak{c}$-spaceable for every $0<p<1$.*

Proof.

In fact, by Theorem 2.1 it follows that $\mathcal{L}\left(\ell_{p};\ell_{p}\right)\diagdown\bigcup\limits_{1\leq s\leq r<\infty}{\textstyle\prod\nolimits_{\left(r,s\right)}}\left(\ell_{p};\ell_{p}\right)$ is non-empty. So, consider the operator $T:\ell_{p}\longrightarrow\mathcal{L}\left(\ell_{p};\ell_{p}\right)$ defined by

[TABLE]

with $u_{i}$ defined in the preliminaries. It follows from [14, Lemma 2.1] that $T$ is  well-defined, linear and injective. Moreover, using [14, Theorem 3.1] we know that

[TABLE]

Therefore, $\overline{T\left(\ell_{p}\right)}$ is a closed infinite-dimensional subspace of $\mathcal{L}\left(\ell_{p};\ell_{p}\right)$. We just have to show that

[TABLE]

Indeed, let $\Psi\in\overline{T\left(\ell_{p}\right)}\diagdown\left\{0\right\}$. Then, there are sequences $\left(a_{i}^{\left(k\right)}\right)_{i=1}^{\infty}\in\ell_{p}\diagdown\left\{0\right\}$ $\left(k\in\mathbb{N}\right)$ such that

[TABLE]

Note that, for each $k\in\mathbb{N}$,

[TABLE]

Then, from (3.1) we have

[TABLE]

In particular, for $x\in\ell_{p}$ arbitrary we get

[TABLE]

Since convergence in $\ell_{p}$ implies coordinatewise convergence, it follows that

[TABLE]

On the other hand, since each operator $u_{i}\in\mathcal{L}\left(\ell_{p};\ell_{p}\right)\diagdown\bigcup\limits_{1\leq s\leq r<\infty}{\textstyle\prod\nolimits_{\left(r,s\right)}}\left(\ell_{p};\ell_{p}\right)$ (it suffices to use (2.1)), for each $i\in\mathbb{N}$, there exist a sequence $\left(x^{\left(j\right)}\right)_{j=1}^{\infty}\in\ell_{s}^{w}\left(\ell_{p}\right)$ such that $\left(u_{i}(x^{\left(j\right)}\right))_{j=1}^{\infty}\notin\ell_{r}\left(\ell_{p}\right)$, that is,

[TABLE]

for each $\varphi\in\left(\ell_{p}\right)^{\prime}=\ell_{\infty}$ because $0<p<1$ (see [10, Theorem 2.3]). So, using (3.2) we get

[TABLE]

Since $\left(\alpha_{i}\right)_{i}\neq 0$ (it follows from the use of the $p$-norm in (3.2)), let $i_{0}$ be such that $\alpha_{i_{0}}\neq 0$. Since the supports of the operators $u_{i}$ are pairwise disjoint for all $i$, from (3.3) we have

[TABLE]

and thus

[TABLE]

We conclude that $\Psi\in\mathcal{L}\left(\ell_{p};\ell_{p}\right)\diagdown\bigcup\limits_{1\leq s\leq r<\infty}{\textstyle\prod\nolimits_{\left(r,s\right)}}\left(\ell_{p};\ell_{p}\right)$. Hence,

[TABLE]

finishing the proof. ∎