Primarity of direct sums of Orlicz spaces and Marcinkiewicz spaces

Abstract

Let $\mathbb {Y}$ be either an Orlicz sequence space or a Marcinkiewicz sequence space. We take advantage of the recent advances in the theory of factorization of the identity carried on by Lechner (Stud Math 248(3):295–319, 2019) to provide conditions on $\mathbb {Y}$ that ensure that, for any $1\le p\le \infty$, the infinite direct sum of $\mathbb {Y}$ in the sense of $\ell _p$ is a primary Banach space. This way, we enlarge the list of Banach spaces that are known to be primary.

1 Introduction

Within his study of operators through which the identity map factors, Lechner [12] introduced the following condition on the coordinate functionals of an unconditional basis of a Banach space.

Definition 1

Let $(\mathbf {x}_j)_{j=1}^\infty$ be an unconditional basis for a Banach space $\mathbb {X}$. We say that its sequence $(\mathbf {x}_j^*)_{j=1}^\infty$ of coordinate functionals is a non-$\ell _1$-splicing weak* basis if for every $A\subseteq \mathbb {N}$ infinite and for every $\theta >0$ there is a sequence $(A_n)_{n=1}^\infty$ consisting of pairwise disjoint infinite subsets of A such that for every $(f_n^*)_{n=1}^\infty$ in $B_{\mathbb {X}^*}$ there is a sequence of scalars $(a_n)_{n=1}^\infty \in S_{\ell _1}$ satisfying

$$\begin{aligned} \left\| \sum _{n=1}^\infty a_n \, P_{A_n}^*(f_n^*)\right\| \le \theta . \end{aligned}$$

We say that $(\mathbf {x}_j^*)_{j=1}^\infty$ is $\ell _1$-splicing if it fails to be non-$\ell _1$-splicing.

Here, and throughout this note, $B_\mathbb {X}$ (respectively $S_\mathbb {X}$) denotes the closed unit ball (resp. unit sphere) of a Banach space $\mathbb {X}$. The symbol $P_A$ denotes the coordinate projection on a set $A\subseteq \mathbb {N}$ with respect to an unconditional basis $\mathcal {B}=(\mathbf {x}_j)_{j=1}^\infty$ of $\mathbb {X}$, i.e., if $\mathcal {B}^*=(\mathbf {x}_j^*)_{j=1}^\infty$ is the sequence of coordinate functionals associated to the basis $\mathcal {B}$, also called the dual basic sequence of $\mathcal {B}$, then $P_A:\mathbb {X}\rightarrow \mathbb {X}$ is defined by

$$\begin{aligned} P_A(f)=\sum _{j\in A} \mathbf {x}_j^*(f)\, \mathbf {x}_j, \quad f\in \mathbb {X}. \end{aligned}$$

(1)

Note that the dual coordinate projection$P_A^*:\mathbb {X}^* \rightarrow \mathbb {X}^*$ of $P_A$ is given by

$$\begin{aligned} P_A^*(f^*) ={{\,\mathrm{w*-}\,}}\sum _{j\in A} f^*(\mathbf {x}_j)\, \mathbf {x}_j^*, \quad f^*\in \mathbb {X}^*. \end{aligned}$$

Since the basis $\mathcal {B}$ is, up to equivalence, univocally determined by the basic sequence $\mathcal {B}^*$ (see [4, Corollary 3.2.4]) it is natural to consider being non-$\ell _1$-splicing as a condition on $\mathcal {B}^*$ instead of as a condition on $\mathcal {B}$.

In the aforementioned paper, Lechner achieved the following contribution to the theory of primary Banach spaces and the factorization of the identity. Recall that a Banach space $\mathbb {X}$ is said to be primary if whenever $\mathbb {Y}$ and $\mathbb {Z}$ are Banach spaces such that $\mathbb {Y}\oplus \mathbb {Z}\approx \mathbb {X}$, then either $\mathbb {Y}\approx \mathbb {X}$ or $\mathbb {Z}\approx \mathbb {X}$. A basis is said to be subsymmetric if it is unconditional and equivalent to all its subsequences. The infinite direct sum of a Banach space $\mathbb {X}$ in the sense of $\ell _p$ (respectively $c_0$) will be denoted by $\ell _p(\mathbb {X})$ (resp. $c_0(\mathbb {X})$). $\mathcal {L}(\mathbb {X})$ will denote the Banach algebra of automorphisms of a Banach space $\mathbb {X}$. We say that the identity map on $\mathbb {X}$ factors through an operator $R\in \mathcal {L}(\mathbb {X})$ if there are operators S and $T\in \mathcal {L}(\mathbb {X})$ such that $T\circ R \circ S =\mathrm {Id}_\mathbb {X}$.

Theorem 1

(see [12, Theorems 1.1 and 1.2]) Suppose that $\mathbb {X}$ is a Banach space equipped with a subsymmetric basis whose dual basic sequence is non-$\ell _1$-splicing. Let $1\le p\le \infty$ and let $\mathbb {Y}$ be either $\mathbb {X}^*$ or $\ell _p(\mathbb {X}^*)$. Then, given $T\in \mathcal {L}(\mathbb {Y})$, the identity map on $\mathbb {Y}$ factors through either T or $\mathrm {Id}_\mathbb {Y}-T$. Consequently, $\ell _p(\mathbb {X}^*)$ is a primary Banach space.

Before undertaking the task of using Theorem 1 for obtaining new primary Banach spaces, we must go over the state-of-the-art on this topic. Casazza et al. [8] proved that if $\mathbb {X}$ has a symmetric basis, i.e., a basis which is equivalent to all its permutations, then the Banach spaces $c_0(\mathbb {X})$ and, in the case when $1< p < \infty$ and $\mathbb {X}$ is not isomorphic to $\ell _1$, $\ell _p(\mathbb {X})$ are primary. Shortly later, Samuel [17] proved that $\ell _p(\ell _r),$$c_0(\ell _r)$ and $\ell _r(c_0)$ are, for $1\le p,r<\infty$, primary Banach spaces. Subsequently, Capon [6] completed the study by proving that $\ell _1(\mathbb {X})$ and $\ell _\infty (\mathbb {X})$ are primary Banach spaces whenever $\mathbb {X}$ possesses a symmetric basis. Symmetric bases are subsymmetric [11, 19], and, in practice, the only information that one needs about symmetric bases in many situations is its subsymmetry. So, it is natural to wonder if the proofs carried on in [6, 8] still work when dealing with subsymmetric bases. A careful look at these papers reveals that it is the case. Summarizing, we have the following result.

Theorem 2

(see [6, 8, 17]) Let $\mathbb {X}$ be a Banach space endowed with a subsymmetric basis. Then $c_0(\mathbb {X})$ and $\ell _p(\mathbb {X})$, $1\le p \le \infty$, are primary Banach spaces.

At this point, we must mention that, as Pełczyński decomposition method is a pivotal tool for facing the study of primary Banach spaces, the task of proving that $\ell _p(\mathbb {X})$ is primary is, in some sense, easier than that of proving that $\mathbb {X}$ is. In fact, as far as we know, $\ell _p$, $1\le p<\infty$, and $c_0$ are the only known primary Banach spaces endowed with a subsymmetric basis.

In light of Theorem 2, applying Theorem 1 to a Banach space $\mathbb {X}$ equipped with a shrinking (subsymmetric) basis does not add a new space to the list of primary Banach spaces. So, taking into account [4, Theorem 3.3.1], within the goal of using Theorem 1 for finding new primary spaces, we must apply it to Banach spaces $\mathbb {X}$ containing a complemented copy of $\ell _1$. Among them, $\mathbb {X}=\ell _1$ seems to be the first space we have to consider. It is timely to bring up the following result.

Theorem 3

(see [8]) Let $1\le p\le \infty$. Then $\ell _p(\ell _\infty )$ is primary.

It is known [12, Proposition 6.2] that the unit vector system of $\ell _\infty$, which is, under the natural pairing, the dual basic sequence of the unit vector system of $\ell _1$, is a non-$\ell _1$-splicing weak* basis. This result combined with Theorem 1 provides an alternative proof to Theorem 3. From an opposite perspective, in order to take advantage of Theorem 3 for obtaining new primary Banach spaces, we need to find weak* bases, other than the unit vector system of $\ell _\infty$, that are non-$\ell _1$-splicing and are not boundedly complete. Within this area of research, Lechner [12] exhibited that the unit vector system of some Orlicz spaces and the dual basis of the unit vector system of some Lorentz sequence spaces fulfil these requirements.

In this manuscript, we go on with the search of non-$\ell _1$-splicing weak* bases and, hence, of new primary Banach spaces, initiated in [12]. In Sect. 4, we generalize [12, Theorem 6.4] by characterizing, in terms of the convex Orlicz M, when the unit vector system of the Orlicz sequence space $\ell _M$ is non-$\ell _1$-splicing. In Sect. 3 we override [12, Theorem 6.5] by describing those weights $\mathbf {s}$ for which the unit vector system of the Marcinkiewicz space $m(\mathbf {s})$ is non-$\ell _1$-splicing. Previously to these sections, in Sect. 2, we carry on a detailed analysis of the concept of non-$\ell _1$-splicing weak* basis introduced by Lechner.

Throughout this article we follow standard Banach space terminology and notation as can be found in [4]. We single out the notation that is more commonly employed. We will denote by $\mathbb {F}$ the real or complex field. By a sign we mean a scalar of modulus one. We denote by $(\mathbf {e}_k)_{k=1}^\infty$ the unit vector system of $\mathbb {F}^\mathbb {N}$, i.e., $\mathbf {e}_k=(\delta _{k,n})_{n=1}^\infty$, were $\delta _{k,n}=1$ if $n=k$ and $\delta _{k,n}=0$ otherwise. The linear span of the unit vector system will be denoted by $c_{00}$.

Given families of non-negative real numbers $(\alpha _i)_{i\in I}$ and $(\beta _i)_{i\in I}$ and a constant $C<\infty$, the symbol $\alpha _i\lesssim _C \beta _i$ for $i\in I$ means that $\alpha _i\le C \beta _i$ for every $i\in I$, while $\alpha _i\approx _C \beta _i$ for $i\in I$ means that $\alpha _i\lesssim _C \beta _i$ and $\beta _i\lesssim _C \alpha _i$ for $i\in I$. A basis will be a Schauder basis. Suppose $(\mathbf {x}_j)_{j=1}^\infty$ and $(\mathbf {y}_j)_{j=1}^\infty$ are bases. We say that $(\mathbf {y}_j)_{j=1}^\infty$C-dominates$(\mathbf {x}_j)_{j=1}^\infty$ (respectively $(\mathbf {y}_j)_{j=1}^\infty$ is C-equivalent to $(\mathbf {x}_j)_{j=1}^\infty$), and write $(\mathbf {x}_j)_{j=1}^\infty \lesssim _C(\mathbf {y}_j)_{j=1}^\infty$ (resp. $(\mathbf {x}_j)_{j=1}^\infty \approx _C(\mathbf {y}_j)_{j=1}^\infty$) if

$$\begin{aligned} \left\| \sum _{j=1}^\infty a_j\mathbf {x}_j\right\| \lesssim _C\left\| \sum _{j=1}^\infty a_j\mathbf {y}_j\right\| \left( \text {resp. } \left\| \sum _{j=1}^\infty a_j\mathbf {x}_j\right\| \approx _C\left\| \sum _{j=1}^\infty a_j\mathbf {y}_j\right\| \right) \end{aligned}$$

for $(a_j)_{j=1}^\infty \in c_{00}$. In all the above cases, when the value of the constant C is irrelevant, we simply drop it from the notation. A basis is said to be unconditional if all its permutations are basic sequences. I turn, we say that a basis $(\mathbf {x}_j)_{j=1}^\infty$ is C-unconditional if $(\mathbf {x}_j)_{n=1}^\infty \approx _C (\epsilon _j\mathbf {x}_j)_{j=1}^\infty$ for any choice of signs $(\epsilon _j)_{j=1}^\infty$. If $(\mathbf {x}_j)_{j=1}^\infty$ is a C-unconditional basis of a Banach space $\mathbb {X}$ and $A\subseteq \mathbb {N}$, then the operator $P_A$ defined as in (1) is well-defined and satisfies $\Vert P_A \Vert \le C$. It is well-known (see e.g. [4, Proposition 3.1.3]) that a basis $\mathcal {B}$ is unconditional if and only if there exists a constant $C\ge 1$ such that $\mathcal {B}$ is C-unconditional.

We say that a sequence in a Banach space is a basic sequence if it is a basis of its closed linear span. If $\mathcal {B}$ is a basis of a Banach space $\mathbb {X}$, then its coordinate functionals constitute a basic sequence in $\mathbb {X}^*$. Reciprocally, if $\mathcal {B}$ is a basis of a Banach space $\mathbb {Y}$ and $\mathbb {X}:=\mathbb {X}[\mathcal {B}]$ is the closed linear span of $\mathcal {B}^*$ in $\mathbb {Y}^*$, then there is a natural isomorphic embedding of $\mathbb {Y}$ into $\mathbb {X}^*:=\mathbb {Y}[\mathcal {B}]$ and, via this embedding, $\mathcal {B}$ is the dual basic sequence of $\mathcal {B}^*$ (see [4, Proposition 3.2.3 and Corollary 3.2.4]). Consequently, any basic sequence is the dual basic sequence of some basis. So, it makes sense to wonder if a given unconditional basic sequence $\mathcal {B}$ of a Banach space (regarded as a sequence in the Banach space $\mathbb {Y}[\mathcal {B}]$ constructed as above) is non-$\ell _1$-splicing.

A basis $\mathcal {B}=(\mathbf {x}_j)_{j=1}^\infty$ of $\mathbb {X}$ is said to be boundedly complete if whenever $(a_j)_{j=1}^\infty \in \mathbb {F}^\mathbb {N}$ satisfies $\sup _n\Vert \sum _{j=1}^n a_j \, \mathbf {x}_j\Vert <\infty$ there is $f\in \mathbb {X}$ such that $\mathbf {x}_j^*(f)=a_j$ for every $j\in \mathbb {N}$. The basis $\mathcal {B}$ is said to be shrinking if $\mathcal {B}^*$ is a basis of the whole space $\mathbb {X}^*$. It is known [10] that a basis $\mathcal {B}$ is boundedly complete if and only if $\mathcal {B}^*$ is shrinking.

The symbol $f={{\,\mathrm{w*-}\,}}\sum _{n=1}^\infty f_n$ means that the series $\sum _{n=1}^\infty f_n$ in $\mathbb {X}^*$ converges to $f\in \mathbb {X}^*$ in the weak* topology of the dual space $\mathbb {X}^*$. Recall that if $\mathcal {B}=(\mathbf {x}_j)_{j=1}^\infty$ is a basis of a Banach space $\mathbb {X}$ and $\mathcal {B}^*(\mathbf {x}_j^*)_{j=1}^\infty$ is its sequence of coordinate functionals, then, for every $f^*\in \mathbb {X}^*$, $(f^*(\mathbf {x}_j))_{j=1}^\infty$ is the unique sequence $(a_j)_{j=1}^\infty \in \mathbb {F}^\mathbb {N}$ such that $f^*={{\,\mathrm{w*-}\,}}\sum _{j=1}^\infty a_j \, \mathbf {x}^*_j$. So, $\mathcal {B}^*$ is a weak* basis of $\mathbb {X}^*$.

The support of a vector $f\in \mathbb {X}$ with respect to the basis $\mathcal {B}$ is the set

$$\begin{aligned} {{\,\mathrm{supp}\,}}(f)=\{j\in \mathbb {N}:\mathbf {x}_j^*(f)\not =0\}, \end{aligned}$$

and the support of a functional $f^*\in \mathbb {X}^*$ with respect to the basis $\mathcal {B}$ is the set

$$\begin{aligned} {{\,\mathrm{supp}\,}}(f^*)=\{j\in \mathbb {N}:f^*(\mathbf {x}_j)\not =0\}. \end{aligned}$$

A sequence $(f_n)_{n=1}^\infty$ in either $\mathbb {X}$ or $\mathbb {X}^*$ is said to be disjointly supported if $({{\,\mathrm{supp}\,}}(f_n))_{n=1}^\infty$ is a sequence of pairwise disjoint subsets of $\mathbb {N}$. A block basic sequence is a sequence $(f_n)_{n=1}^\infty$ for which there is an increasing sequence $(k_n)_{n=1}^\infty$ of positive integers such that, with the convention $n_0=0$, ${{\,\mathrm{supp}\,}}(f_n)\subseteq [1+k_{n-1},k_n]$ for every $n\in \mathbb {N}$. Block basic sequences are a particular case of disjointly supported sequences. Since any block basic sequence is a basic sequence, our terminology is consistent. Note that any disjointly supported sequence (in either $\mathbb {X}$ or $\mathbb {X}^*$) with respect to an unconditional basis of a Banach space $\mathbb {X}$ is an unconditional basic sequence.

Let $\mathbb {X}\subseteq \mathbb {F}^\mathbb {N}$ be a Banach space for which the unit vector system is a basis. We say that a Banach space $\mathbb {Y}\subseteq \mathbb {F}^\mathbb {N}$ is the dual space of$\mathbb {X}$under the natural pairing if there is an isomorphism $T:\mathbb {X}^*\rightarrow \mathbb {Y}$ such that $T(f)(g)=\sum _{j=1}^\infty a_j b_j$ for every $f=(a_j)_{j=1}^\infty \in \mathbb {Y}$ and every $g=(b_j)_{j=1}^\infty \in c_{00}$. Observe that if $\mathbb {Y}$ is, under the natural pairing, the dual space of $\mathbb {X}$ and $f=(a_j)_{j=1}^\infty$ belongs to either $\mathbb {X}$ or $\mathbb {Y}$, then the support of f with respect to the unit vector system is the set ${{\,\mathrm{supp}\,}}(f)=\{j \in \mathbb {N}:a_j\not =0\}$.

A sequence $=(f_j)_{j=1}^\infty$ is a Banach space is said to be semi-normalized if $\inf _j\Vert f_j \Vert >0$ and $\sup _j \Vert f_j \Vert <\infty$. Note that subsymmetric bases are semi-normalized.

Other more specific notation will be specified in context when needed.

2 Non-$\ell _1$-splicing weak* bases

The main goal of the study carried on in this section is to show that, if the basis is subsymmetric, we can describe more simply non-$\ell _1$-splicing bases. In order to prove our results, it will be convenient to introduce some additional terminology.

If $\mathcal {B}=(\mathbf {x}_j)_{j=1}^\infty$ is a subsymmetric basis of a Banach space $\mathbb {X}$ then [5, Theorem 3.7] there is a renorming of $\mathbb {X}$ with respect to which it is 1-subsymmetric, i.e., $\mathcal {B}$ is 1-unconditional and for every increasing map $\phi :\mathbb {N}\rightarrow \mathbb {N}$ the linear operator

$$\begin{aligned} V_\phi :\mathbb {X}\rightarrow \mathbb {X},\quad \sum _{j=1}^\infty a_j \, \mathbf {x}_j \mapsto \sum _{j=1}^\infty a_j \, \mathbf {x}_{\phi (j)} \end{aligned}$$

(2)

is an isometric embedding. If the basis is 1-subsymmetric then the linear operator

$$\begin{aligned} U_\phi :\mathbb {X}\rightarrow \mathbb {X},\quad \sum _{j=1}^\infty a_j \, \mathbf {x}_j \mapsto \sum _{j=1}^\infty a_{\phi (j)}\, \mathbf {x}_{j} \end{aligned}$$

(3)

is norm-one for every increasing map $\phi :\mathbb {N}\rightarrow \mathbb {N}$ (see e.g. [5, Lemma 3.3]). The dual operators of $V_\phi$ and $U_\phi$ are given by

$$\begin{aligned} V_\phi ^*&:\mathbb {X}^*\rightarrow \mathbb {X}^*,\quad {{\,\mathrm{w*-}\,}}\sum _{j=1}^\infty a_j \, \mathbf {x}_j^* \mapsto {{\,\mathrm{w*-}\,}}\sum _{j=1}^\infty a_{\phi (j)}\, \mathbf {x}_{j}^*,\\ U_\phi ^*&:\mathbb {X}^*\rightarrow \mathbb {X}^*,\quad {{\,\mathrm{w*-}\,}}\sum _{j=1}^\infty a_j \, \mathbf {x}_j^* \mapsto {{\,\mathrm{w*-}\,}}\sum _{j=1}^\infty a_j \, \mathbf {x}_{\phi (j)}^*. \end{aligned}$$

Since $U_\phi \circ V_\phi =\mathrm {Id}_\mathbb {X}$, we have $V_\phi ^*\circ U_\phi ^*=\mathrm {Id}_{\mathbb {X}^*}$. Consequently, $U_\phi ^*$ is an isomorphic embedding (isometric embedding if $\mathcal {B}$ is 1-subsymmetric).

With this background in our hands, we are almost ready to prove the aforementioned characterization of non-$\ell _1$-splicing weak* subsymmetric bases. Before doing so, we bring up a result that is implicit in [12].

Lemma 1

(cf. [12, Proposition 6.1]) Let $\mathcal {B}$ be an unconditional basis of a Banach space. Assume that $\mathcal {B}^*$ is $\ell _1$-splicing. Then there is a sequence of disjointly supported functionals in $\mathbb {X}^*$ equivalent to the unit vector system of $\ell _1$.

Proof

Our hypothesis says that there are $\theta >0$ and $A\subseteq \mathbb {N}$ infinite such that, for every sequence $(A_n)_{n=1}^\infty$ consisting of pairwise disjoint infinite subsets of A, there is $(f_n^*)_{n=1}^\infty$ in $B_{\mathbb {X}^*}$ with $\theta <\Vert \sum _{n=1}^\infty a_n \, P_{A_n}^*(f_n^*)\Vert$ for every $(a_n)_{n=1}^\infty \in S_{\ell _1}$.

Pick out an arbitrary sequence $(A_n)_{n=1}^\infty$ consisting of pairwise disjoint infinite subsets of A and let $(f_n^*)_{n=1}^\infty$ be as above. If $g_n^*= P_{A_n}^*(f_n^*)$, we have ${{\,\mathrm{supp}\,}}(g_n^*)\subseteq A_n$ and $\sup _n \Vert g_n^*\Vert <\infty$ for every $n\in \mathbb {N}$. We infer that the disjointly supported sequence $(g_n^*)_{n=1}^\infty$ is equivalent to the unit vector basis of $\ell _1$. $\square$

For reference, we write down the following elementary lemma, which we will use in the proof of the subsequent theorem.

Lemma 2

Let $(B_n)_{n=1}^\infty$ be a sequence of disjointly supported subsets of $\mathbb {N}$ and let $(A_n)_{n=1}^\infty$ be a sequence of disjointly supported infinite subsets of $\mathbb {N}$. Then, there exists an increasing map $\phi :\mathbb {N}\rightarrow \mathbb {N}$ such that $\phi (B_n)\subseteq A_n$ for every $n\in \mathbb {N}$.

Proof

Clearly, it suffices to prove the result in the case when $(B_n)_{n=1}^\infty$ is a partition of $\mathbb {N}$. Define $\nu :\mathbb {N}\rightarrow \mathbb {N}$ by $\nu (k)=n$ if $k\in B_n$. By hypothesis,

$$\begin{aligned} D_{n,m}:=\{ j \in A_n :j>m\} \end{aligned}$$

is non-empty for every $n\in \mathbb {N}$ and every $m\in \mathbb {N}\cup \{0\}$. With the convention $\phi (0)=0$, we define $\phi :\mathbb {N}\rightarrow \mathbb {N}$ by means of the recursive formula

$$\begin{aligned} \phi (k)=\min D_{\nu (k),\phi (k-1)}, \quad k\in \mathbb {N}. \end{aligned}$$

It is clear that $\phi$ satisfies the desired properties. $\square$

Theorem 4

Assume that $\mathcal {B}$ is a subsymmetric basis of a Banach space $\mathbb {X}$. Then its dual basic sequence $\mathcal {B}^*$ is $\ell _1$-splicing if and only there is a sequence of disjointly supported functionals in $\mathbb {X}^*$ equivalent to the unit vector system of $\ell _1$.

Proof

The “only if” part follows from Lemma 1. Assume that there is a disjointly supported sequence $(f_n^*)_{n=1}^\infty$ in $\mathbb {X}^*$ that is equivalent to the unit vector system of $\ell _1$. By dilation, we can assume that $\Vert f_n^*\Vert \le 1$ for every $n\in \mathbb {N}$. Let $c>0$ be such that

$$\begin{aligned} c\sum _{n=1}^\infty |a_n| \le \left\| \sum _{n=1}^\infty a_n \, f_n \right\| , \quad (a_n)_{n=1}^\infty \in \ell _1. \end{aligned}$$

We also assume, without loss of generality, that $\mathcal {B}$ is 1-subsymmetric. Choose $0<\theta <c$ and $A=\mathbb {N}$. Pick a sequence $(A_n)_{n=1}^\infty$ consisting of pairwise disjoint infinite subsets of $\mathbb {N}$. By Lemma 2, there is an increasing map $\phi :\mathbb {N}\rightarrow \mathbb {N}$ such that $\phi ({{\,\mathrm{supp}\,}}(f_n^*))\subseteq A_n$. Put $g_n^*=U_\phi ^*(f_n^*)$ for $n\in \mathbb {N}$. Then, taking into account that $U_\phi ^*$ is an isometric embedding, we have $P_{A_n}^*(g_n^*)=g_n^*\in B_{\mathbb {X}^*}$ for every $n\in \mathbb {N}$, and

$$\begin{aligned} \theta <c\le \left\| \sum _{n=1}^\infty a_n \, g_n^* \right\| \end{aligned}$$

for every $(a_n)_{n=1}^\infty \in S_{\ell _1}$. Consequently, $\mathcal {B}^*$ is $\ell _1$-splicing. $\square$

Note that, if $\mathbb {X}$ has an unconditional basis and $\mathbb {X}^*$ is non-separable, then $\ell _1$ is a subspace of $\mathbb {X}^*$ (see [4, Theorems 2.5.7 and 3.3.1]). So, Theorems 1 and 4 reveal that the position in which $\ell _1$ is (and is not) placed inside $\mathbb {X}^*$ has significative structural consequences.

Next, we give some consequences of Theorem 4. First of them was previously achieved in [12].

Lemma 3

Let $\mathcal {B}=(f_n)_{n=1}^\infty$ be a semi-normalized disjointly supported sequence in $\ell _\infty$. Then $\mathcal {B}$ is equivalent to the unit vector system of $\ell _\infty$.

Proof

Denote $c=\inf _n\Vert f_n \Vert$ and $C=\sup _n \Vert f_n \Vert$. It is clear that $c \Vert g \Vert _\infty \le \Vert \sum _{n=1}^\infty a_n f_n\Vert \le C \Vert g \Vert _\infty$ for every $g=(a_n)_{n=1}^\infty \in c_{00}$. $\square$

Proposition 1

(see [12, Proposition 6.2]) The unit vector system of $\ell _\infty$ is non-$\ell _1$-splicing.

Proof

The unit vector system of $\ell _\infty$, denoted by $\mathcal {B}_\infty$ in this proof, is, under the natural pairing, the dual basic sequence of the unit vector basis of $\ell _1$, denoted by $\mathcal {B}_1$. Suppose by contradiction that there is a disjointly supported sequence $\mathcal {B}$ in $\ell _\infty$ that is equivalent to $\mathcal {B}_1$. Then, in particular, $\mathcal {B}$ is semi-normalized. Invoking Lemma 3 we obtain $\mathcal {B}_1\approx \mathcal {B}\approx \mathcal {B}_\infty$. This absurdity, combined with Theorem 4, proves that $\mathcal {B}_\infty$ f is non-$\ell _1$-splicing. $\square$

Proposition 2

Let $\mathcal {B}$ be a subsymmetric basis of a Banach space $\mathbb {X}$ whose dual basic sequence $\mathcal {B}^*$ is non-$\ell _1$-splicing. Then $\mathcal {B}$ is boundedly complete and $\mathcal {B}^*$ is shrinking.

Proof

If $\mathcal {B}^*$ fails to be shrinking, then, by [4, Theorem 3.3.1], there is a block basic sequence with respect to $\mathcal {B}^*$ equivalent to the unit vector system of $\ell _1$. Consequently, by Theorem 4, $\mathcal {B}^*$ is $\ell _1$-splicing. We complete the proof by appealing to [4, Theorem 3.2.15]. $\square$

Corollary 1

Let $\mathbb {X}$ be a Banach space endowed with a subsymmetric basis $\mathcal {B}$. If $\mathcal {B}^*$ is non-$\ell _1$-splicing, then $\mathbb {X}$ is a dual space (and $\mathbb {X}^*$ is a bidual space).

Proof

It is immediate from combining Proposition 2 with [4, Theorem 3.2.15]. $\square$

Corollary 2

Let $\mathbb {X}$ be a Banach space endowed with a subsymmetric shrinking basis $\mathcal {B}$. If $\mathcal {B}^*$ is non-$\ell _1$-splicing, then $\mathbb {X}$ is reflexive.

Proof

Just combine Proposition 2 with [4, Theorem 3.2.19].$\square$

Proposition 3

Let $\mathbb {X}$ be a reflexive Banach space endowed with a subsymmetric basis $\mathcal {B}$. Then $\mathcal {B}$ and $\mathcal {B}^*$ are non-$\ell _1$-splicing.

Proof

By [4, Theorems 3.2.15, 3.2.19 and 3.3.1], neither $\mathbb {X}$ nor $\mathbb {X}^*$ contain a subspace isomorphic to $\ell _1$. Then, by Theorem 4, $\mathcal {B}$ and $\mathcal {B}^*$ are non-$\ell _1$-splicing. $\square$

3 Marcinkiewicz spaces

A weight will be a sequence of positive scalars. Given a weight $\mathbf {s}=(s_j)_{j=1}^\infty$ the Marcinkiewicz space$m(\mathbf {s})$ is the set consisting of all sequences $f=(a_j)_{j=1}^\infty \in \mathbb {F}^\mathbb {N}$ such that

$$\begin{aligned} \Vert f\Vert _{m(\mathbf {s})}:= \sup \left\{ \frac{1}{s_n} \sum _{j\in A} |a_j| :n\in \mathbb {N}, \, |A|=n\right\} <\infty . \end{aligned}$$

It is clear, and well-known, that $(m(\mathbf {s}), \Vert \cdot \Vert _{m(\mathbf {s})})$ is a Banach space and that the unit vector system is a symmetric basic sequence in $m(\mathbf {s})$. If $f\in c_0$ and $(a_n^*)_{n=1}^\infty$ denotes its non-increasing rearrangement then

$$\begin{aligned} \Vert f\Vert _{m(\mathbf {s})}=\sup _n \frac{1}{s_n} \sum _{k=1}^n a_k^*. \end{aligned}$$

(4)

It is not hard to prove that if $\lim _n {s_n}/{n}=0$ then $m(\mathbf {s})\subseteq c_0$ continuously. Otherwise, we have $m(\mathbf {s})=\ell _\infty$ (up to an equivalent norm).

The next proposition gathers some results that relate Marcinkiewicz spaces to Lorentz spaces. Prior to enunciate it, let us fix some terminology. A weight $(w_j)_{j=1}^\infty$ is said to be regular if it satisfies the Dini condition

$$\begin{aligned} \sup _n \frac{1}{n w_n} \sum _{j=1}^n w_j <\infty . \end{aligned}$$

If two weights $\mathbf {w}=(w_j)_{j=1}^\infty$ and $\mathbf {s}=(s_n)_{n=1}^\infty$ are related by the formula

$$\begin{aligned} s_n=\sum _{j=1}^n w_j, \quad n\in \mathbb {N}, \end{aligned}$$

we say that $\mathbf {s}$ is the primitive weight of $\mathbf {w}$ and that $\mathbf {w}$ is the discrete derivative of $\mathbf {s}$. Given a weight $\mathbf {w}=(w_j)_{j=1}^\infty$ with primitive weight $\mathbf {s}=(s_j)_{j=1}^\infty$, the Lorentz space $d(\mathbf {w},1)$ (respectively weak Lorentz space $d(\mathbf {w},\infty )$) is the set consisting of all sequences $f \in c_0$ whose non-increasing rearrangement $(a_j^*)_{j=1}^\infty$ fulfils

$$\begin{aligned} \Vert f \Vert _{d(\mathbf {w},1)}:=\sum _{j=1}^\infty a_j^* w_j<\infty \left( \text {resp. }\Vert f \Vert _{d(\mathbf {w},\infty )}:=\sup _{j\in \mathbb {N}} a_j^*s_j<\infty \right) . \end{aligned}$$

If $(s_n/n)_{n=1}^\infty$ is non-increasing, then $d(\mathbf {w},1)$ is a Banach space (see [7, Theorem 2.5.10]). In turn, if $\mathbf {s}$ is doubling, then $d(\mathbf {w},\infty )$ is a quasi-Banach space (see [7, Theorem 2.2.16]). It is not hard to prove that $c_{00}$ is a dense subspace of $d(\mathbf {w},1)$. Then, the unit vector system is a symmetric basis of $d(\mathbf {w},1)$.

Proposition 4

(See [7, Theorems 2.4.14 and 2.5.10 and Corollary 2.4.26]; see also [2, Section 6]) Let $\mathbf {w}=(w_j)_{j=1}^\infty$ be a decreasing weight with $\lim _j w_j=0$, let $\mathbf {s}$ denote its primitive weight, and let $\mathbf {v}$ denote the discrete derivative of the inverse weight $\mathbf {w}^{-1}=(1/w_j)_{j=1}^\infty$ of $\mathbf {w}$.

• (i) $m(\mathbf {s})$ is, under the natural pairing, the dual space of $d(\mathbf {w},1)$.

• (ii) If $\mathbf {w}$ is a regular weight, then $d(\mathbf {v},\infty )=m(\mathbf {s})$ (up to an equivalent quasi-norm).

In route to state the main result of the section, we introduce some additional conditions on weights, and we bring up a result involving them. We say that a weight $(s_n)_{n=1}^\infty$ is essentially decreasing (respectively essentially increasing) if

$$\begin{aligned} \inf _{m\le n} \frac{s_m}{s_n}>0 \left( \text {resp. } \sup _{m\le n} \frac{s_m}{s_n}<\infty \right) . \end{aligned}$$

Note that $(s_n)_{n=1}^\infty$ is essentially decreasing (resp. essentially increasing) if and only if it is equivalent to a non-increasing (resp. non-decreasing) weight. We say that a weight $(s_n)_{n=1}^\infty$ has the lower regularity property (LRP for short) if there is a constant $C>1$ and an integer $r\ge 2$ such that

$$\begin{aligned} s_{rn}\ge C s_n, \quad n\in \mathbb {N}. \end{aligned}$$

Lemma 4

(see [1, Lemma 2.12]) Let $\mathbf {s}=(s_n)_{n=1}^\infty$ be a essentially increasing weight such that $\mathbf {w}=(s_n/n)_{n=1}^\infty$ is essentially decreasing. Then, $\mathbf {s}$ has the LRP if and only if $\mathbf {w}$ is a regular weight.

The following result is rather straightforward, and old-timers will surely be aware of it and could produce its proof on the spot. Nonetheless, for later reference and expository ease, we record it.

Lemma 5

Let $\mathcal {B}=(\mathbf {x}_j)_{j=1}^\infty$ be a subsymmetric basis of a Banach space $\mathbb {X}$ such that $n\lesssim \Vert \sum _{j=1}^n \mathbf {x}_j \Vert$ for $n\in \mathbb {N}$. Then $\mathcal {B}$ is equivalent to the unit vector basis of $\ell _1$.

Proof

By [15, Proposition 3.a.4], for $f= \sum _{j=1}^\infty a_j \, \mathbf {x}_j \in \mathbb {X}$ and $n\in \mathbb {N}$ we have

$$\begin{aligned} \left| \sum _{j=1}^n a_j \right| \lesssim \left| \frac{1}{n} \sum _{j=1}^n a_j \right| \, \left\| \sum _{j=1}^n \mathbf {x}_j \right\| \lesssim \left\| f \right\| . \end{aligned}$$

Combining this inequality (which using the terminology introduced by Singer [18] says that $\mathcal {B}$ is a basis of type $P^*$) with unconditionality yields the desired result.$\square$

We are ready to state and prove the main theorem of the present section.

Theorem 5

Let $\mathbf {s}=(s_n)_{n=1}^\infty$ be an increasing weight whose discrete derivative is essentially decreasing. Then, the unit vector system of $m(\mathbf {s})$ is non-$\ell _1$-splicing if and only if

$$\begin{aligned} S:=\inf _{n\in \mathbb {N}} \sup _{k\in \mathbb {N}} \frac{s_k}{s_{kn}}=0. \end{aligned}$$

(5)

Proof

We infer from our assumptions on $\mathbf {s}$ that there is a non-increasing weight $\mathbf {w}=(w_n)_{n=1}^\infty$ whose primitive weight $\mathbf {t}=(t_n)_{n=1}^\infty$ is equivalent to $\mathbf {s}$. So, $m(\mathbf {t})=m(\mathbf {s})$. Moreover, by Lemma 4, if $\mathbf {s}$ had the LRP, we could choose $\mathbf {w}$ to be regular.

If $\lim _j w_j>0$ we would have $s_n \approx n$ for $n\in \mathbb {N}$ and, then, $S=0$. We would also have $m(\mathbf {s})=\ell _\infty$. Therefore, by Proposition 1, $m(\mathbf {s})$ would be non-$\ell _1$-splicing. So, we assume from now on that $\lim _j w_j=0$. Then, by Proposition 4 (i), the unit vector system of $m(\mathbf {t})$ is the dual basic sequence of the unit vector basis of $d(\mathbf {w},1)$.

Given a bijection $\pi :\mathbb {N}^2\rightarrow \mathbb {N}$ we define a disjointly supported sequence $\mathcal {B}_\pi =(f_n)_{n=1}^\infty$ in $\mathbb {F}^\mathbb {N}$ by

$$\begin{aligned} f_n=(a_{j,n})_{j=1}^\infty , \quad a_{j,n}={\left\{ \begin{array}{ll} w_i &{}\quad \text { if } j=\pi (i,n), \\ 0 &{}\quad \text { otherwise.}\end{array}\right. } \end{aligned}$$

The non-increasing rearrangement of each sequence $f_n$ is the sequence $(w_i)_{i=1}^\infty$. Then, by (4), $\Vert f_n \Vert _{m(\mathbf {t})}=1$. We infer that $\mathcal {B}_\pi$ is a symmetric basic sequence in $m(\mathbf {t})$. Given $m\in \mathbb {N}$ the non-increasing rearrangement of $\sum _{n=1}^m f_n$ is the sequence

$$\begin{aligned} (\underbrace{w_1,\dots ,w_1}_{m},\dots , \underbrace{w_k,\dots ,w_k}_{m},\dots ). \end{aligned}$$

Therefore, applying (4) and taking into account that $\mathbf {w}$ is non-increasing,

$$\begin{aligned} \left\| \sum _{n=1}^m f_n \right\| _{m(\mathbf {t})}= & {} \sup _{\begin{array}{c} k \ge 1 \\ 1 \le r \le m \end{array}}\frac{ r w_k +m \sum _{i=1}^{k-1}w_i}{\sum _{i=1}^{m(k-1)+r} w_i}\\\le & {} \sup _{\begin{array}{c} k \ge 1 \\ 1 \le r \le n \end{array}} a_{k,r}^{(m)}, \end{aligned}$$

where

$$\begin{aligned} a_{k,r}^{(m)}=\frac{ r w_k +m \sum _{i=1}^{k-1}w_i}{ \frac{r}{m}\sum _{i=1+m(k-1)}^{mk} w_i+\sum _{i=1}^{m(k-1)} w_i}. \end{aligned}$$

Since, for any $a,b,c,d\in (0,\infty )$, the mapping $t\mapsto (a+bt)/(c+dt)$ is monotone in $(0,\infty )$ we have $a_{k,r}^{(m)}\le \max \{a_{k,0}^{(m)},a_{k,m}^{(m)}\}$ whenever $1\le r \le m$ and $k\ge 2$. Put

$$\begin{aligned} b_k^{(m)}=\frac{ \sum _{i=1}^{k}w_i}{ \sum _{i=1}^{mk} w_i}, \, k,m\in \mathbb {N}, \quad B^{(m)}=\sup _{k\in \mathbb {N}} b_k^{(m)}, \, m\in \mathbb {N}, \quad B=\inf _{m\in \mathbb {N}} B_m. \end{aligned}$$

Note that $a_{k,m}^{(m)} = a_{k+1,0}^{(m)} = m b_{k}^{(m)}$ for every $k\in \mathbb {N}$, and that $a_{1,r}^{(m)}=m b_1^{(m)}$ for every $r\in \mathbb {N}$. Consequently,

$$\begin{aligned} \left\| \sum _{n=1}^m f_n \right\| _{m(\mathbf {t})}= m B^{(m)}, \quad m\in \mathbb {N}. \end{aligned}$$

(6)

Taking into account Theorem 4, we have to prove that $B>0$ if and only if $m(\mathbf {t})$ contains a disjointly supported sequence equivalent to the unit vector basis of $\ell _1$.

Assume that $B>0$. Pick a bijection $\pi$ from $\mathbb {N}^2$ onto $\mathbb {N}$ and let $\mathcal {B}_\pi =(f_n)_{n=1}^\infty$. By (6), $B m \le \Vert \sum _{n=1}^m f_n \Vert _{m(\mathbf {t})}$ for every $m\in \mathbb {N}$. Then, by Lemma 5, $\mathcal {B}_\pi$, regarded as a sequence in $m(\mathbf {t})$, is equivalent to the unit vector system of $\ell _1$.

Reciprocally, assume that $B=0$. In particular, there is $n\in \mathbb {N}$ such that $s_k/s_{nk}\le 1/2$ for every $k\in \mathbb {N}$. Then, $\mathbf {s}$ has the LRP and, consequently, we can, and we do, assume that the weight $\mathbf {w}$ above chosen is regular. Therefore, by Part (ii) of Proposition 4,

$$\begin{aligned} m(\mathbf {s})=m(\mathbf {t})=d(\mathbf {v},\infty ), \end{aligned}$$

where $\mathbf {v}$ is the discrete derivative of. $\mathbf {w}^{-1}$. Let $(g_n)_{n=1}^\infty$ be a disjointly supported sequence in $\mathbb {F}^\mathbb {N}$ with $\sup _n \Vert g_n \Vert _{d(\mathbf {v},\infty )}<\infty$. By the very definition of the quasi-norm in $d(\mathbf {v},\infty )$, there is a bijection $\pi :\mathbb {N}^2\rightarrow \mathbb {N}$ such that $(g_n)_{n=1}^\infty \lesssim \mathcal {B}_\pi$. Consequently, by (6),

$$\begin{aligned} \left\| \sum _{n=1}^m g_n \right\| _{d(\mathbf {v},\infty )}\lesssim m B^{(m)}, \quad m\in \mathbb {N}. \end{aligned}$$

We infer that $\inf _m m^{-1} \Vert \sum _{n=1}^m g_n \Vert _{d(\mathbf {v},\infty )}=0$. Then, $(g_n)_{n=1}^\infty$, regarded as a sequence in $d(\mathbf {v},\infty )$, is not equivalent to the unit vector system of $\ell _1$. $\square$

Remark 1

Suppose that $\mathbf {w}=(w_n)_{n=1}^\infty$ in $c_0\setminus \ell _1$ is non-increasing and fulfils

$$\begin{aligned} \lim _n \sup _k \frac{w_k}{n w_{kn}}=0. \end{aligned}$$

(7)

Lechner [12] proved that the dual basis of the unit vector system of the sequence Lorentz space $d(\mathbf {w},1)$ is non-$\ell _1$-splicing. Consequently, in light of Proposition 4 (i) and Theorem 5, the primitive weight $(s_n)_{n=1}^\infty$ of $\mathbf {w}$ satisfies (5). Let us write down a direct proof for this fact. Pick $\varepsilon >0$. There is $n\in \mathbb {N}$ such that $w_i\le \varepsilon n w_{in}$ for every $i\in \mathbb {N}$. Then, if

$$\begin{aligned} t_i=\sum _{j=in-n+1}^{in} w_j, \end{aligned}$$

we have $w_i\le \varepsilon t_i$ for every $i\in \mathbb {N}$. For all $k\in \mathbb {N}$ we obtain

$$\begin{aligned} \frac{s_k}{s_{kn}}=\frac{\sum _{i=1}^k w_i}{\sum _{i=1}^k t_i} \le \frac{\sum _{i=1}^k \varepsilon t_i}{\sum _{i=1}^k t_i}=\varepsilon . \end{aligned}$$

We emphasize that the converse “almost” holds. Indeed, if $\mathbf {w}\in c_0$ is essentially decreasing and its primitive weight $\mathbf {s}=(s_n)_{n=1}^\infty$ fulfils (5), the proof of Theorem 5 gives a non-increasing regular weight $\mathbf {w}'$ whose primitive weight is equivalent to $\mathbf {s}$. Then $d(\mathbf {w},1)=d(\mathbf {w}',1)$, and $\mathbf {w}'$ satisfies (7).

To give relevance to Theorem 5 we make the effort of telling apart Marcinkiewicz spaces from $\ell _\infty$.

Proposition 5

Let $\mathbf {s}=(s_n)_{n=1}^\infty$ be an increasing weight whose discrete derivative is essentially decreasing. If $\lim _n s_n/n=0$ then $m(\mathbf {s})$ is not an $\mathcal {L}_\infty$-space. In particular, $m(\mathbf {s})$ is not isomorphic to $\ell _\infty$.

Proof

Pick $\mathbf {w}\in \mathcal {W}$ whose primitive weight is equivalent to $\mathbf {s}$. Suppose that $m(\mathbf {s})$ is an $\mathcal {L}_\infty$-space. Then, by Proposition 4 (i) and [14, Theorem III], $d(\mathbf {w},1)$ is an $\mathcal {L}_1$-space. Since the unit vector system is an unconditional basis of $d(\mathbf {w},1)$, taking into account [13, Theorem 6.1], we reach the absurdity $d(\mathbf {w},1)=\ell _1$. $\square$

We close this section by writing down the result that arises from combining Theorem 5 with Theorem 1.

Corollary 3

Let $p\in [1,\infty ]$ and let $\mathbf {s}=(s_n)_{n=1}^\infty$ be an increasing weight whose discrete derivative is essentially decreasing and assume that (5) holds. Let $\mathbb {Y}$ be either $m(\mathbf {s})$ or $\ell _p(m(\mathbf {s}))$. Then, if $T\in \mathcal {L}(\mathbb {Y})$, the identity map on $\mathbb {Y}$ factors through either T or $\mathrm {Id}_\mathbb {Y}-T$. Consequently, $\ell _p(m(\mathbf {s}))$ is a primary Banach space.

4 Orlicz sequence spaces

Throughout this section we follow the terminology on Orlicz spaces and Musielak-Orlicz spaces used in the handbooks [15, 16]. A normalized convex Orlicz function is a convex function $M:[0,\infty )\rightarrow [0,\infty )$ such that $M(0)=0$ and $M(1)=1$. If M vanishes on a neighborhood of the origin, M is said to be degenerate. Given a sequence $\mathbf {M}=(M_n)_{n=1}^\infty$ of normalized convex Orlicz functions, the Musielak–Orlicz norm $\Vert \cdot \Vert _{\ell _\mathbf {M}}$ is the Luxemburg norm built from the modular

$$\begin{aligned} m_\mathbf {M}:\mathbb {F}^\mathbb {N}\rightarrow [0,\infty ], \quad (a_n)_{n=1}^\infty \mapsto \sum _{n=1}^\infty M_n(|a_n|), \end{aligned}$$

that is, $\Vert f \Vert _{\ell _\mathbf {M}}=\inf \{ t >0:m_\mathbf {M}(f/t)\le 1\}$ for all $f\in \mathbb {F}^\mathbb {N}$. The Musielak–Orlicz space $\ell _\mathbf {M}$ is the Banach space consisting of all sequences f for which $\Vert f \Vert _{\ell _\mathbf {M}}<\infty$. Orlicz sequence spaces can be obtained as a particular case of Musielak–Orlicz sequence spaces. Namely, if M is a normalized convex Orlicz functions, we put $\ell _M=\ell _\mathbf {M}$, where, if $\mathbf {M}=(M_n)_{n=1}^\infty$, $M_n=M$ for every $n\in \mathbb {N}$. We will denote by $h_M$ the closed linear span of the unit vector system of $\ell _M$. It is known (see [15, Proposition 4.a.2]) that

$$\begin{aligned} h_M=\{ f \in \mathbb {F}^\mathbb {N}:\forall s<\infty ,\, m_M(sf)<\infty \}. \end{aligned}$$

(8)

It is clear that the unit vector system is a symmetric basic sequence of $\ell _M$ for every normalized Orlicz function M. If $M^*$ is the Orlicz function complementary to M we have

$$\begin{aligned} (h_{M^*})^*=\ell _M \end{aligned}$$

(9)

under the natural pairing (see [15, Proposition 4.b.1]). So, the unit vector system is a basis of $h_M$ and a weak* basis of $\ell _M$.

Let us bring up the following result that we will need.

Theorem 6

(see [16, Theorem 8.11]) Let $\mathbf {M}=(M_n)_{n=1}^\infty$ and $\mathbf {N}=(N_n)_{n=1}^\infty$ be sequences of normalized convex Orlicz functions. Then $\ell _\mathbf {N}\subseteq \ell _\mathbf {M}$ if and only if there are a positive sequence $(a_n)_{n=1}^\infty \in \ell _1$, $\delta >0$, C and $D\in (0,\infty )$ such that

$$\begin{aligned} N_n(t) < \delta \Longrightarrow M_n(t) \le C N_n(D t) + a_n. \end{aligned}$$

Remark 2

Theorem 6 gives, in particular, that $\ell _M=\ell _N$ if and only if there are $a,b>0$ such that $M(t)\approx N(bt)$ for $0\le t\le a$ (see also [15, Proposition 4.a.5]).

If we denote, for $b\in (0,\infty )$,

$$\begin{aligned} M_b(t)=\frac{M(bt)}{M(b)}, \quad t \ge 0, \end{aligned}$$

the indexes $\alpha _M$ and $\beta _M$ of the non-degenerate normalized convex Orlicz function M are defined by $\alpha _M=\sup A_M$ and, with the convention $\inf \emptyset =\infty$, $\beta _M=\inf B_M$, where

$$\begin{aligned} A_M&=\left\{ q\in [1,\infty ) :\sup _{0\le b,t \le 1} t^{-q} M_b(t) <\infty \right\} , \\ B_M&=\left\{ q\in [1,\infty ) :\inf _{0\le b,t \le 1} t^{-q} M_b(t)>0\right\} . \end{aligned}$$

Note that, by convexity, $M(bt)\le t M(b)$ for every $t\in [0,1]$ and $b\in [0,\infty )$. Consequently, $1\in A_M$ and, hence, $\alpha _M$ is well-defined. Note also that, if $q\in A_M\cap B_M$, then $q=\max A_M=\min B_M$. We infer that $1\le \alpha _M\le \beta _M\le \infty$.

Our characterization of Orlicz sequence spaces whose unit vector system is non-$\ell _1$-splicing will be a consequence of the following result.

Theorem 7

(cf. [15, Theorem 4.a.9]) Let M be a non-degenerate normalized convex Orlicz function and $1\le p\le \infty$. The following are equivalent.

• (i) $p\in [\alpha _M,\beta _M]$.

• (ii) There is a disjointly supported sequence with respect to the unit vector system of $\ell _M$ which is equivalent to the unit vector system of $\ell _p$.

• (iii) There is a block basic sequence with respect to the unit vector system of $\ell _M$ which is equivalent to the unit vector system of $\ell _p$.

We emphasize that the equivalence between items (i) and (iii) can be easily obtained from [15, Theorem 4.a.9]. Indeed, it follows from combining [3, Proposition 2.14], [4, Theorem 3.3.1] and Bessaga–Pelczyński Selection Principle that, if a Banach space equipped with an unconditional basis $\mathcal {U}$ contains a subsymmetric basic sequence $\mathcal {B}$, then there is a block basic sequence with respect to $\mathcal {U}$ that is equivalent to $\mathcal {B}$. As it is obvious that (iii) implies (ii), our contribution to the theory of sequence Orlicz spaces consists in proving that (ii) implies (i). Nonetheless, for expository ease, we will put in order all the arguments that come into play in the proof of Theorem 7. We start by writing down some terminology and claims from [15].

Given a non-degenerate normalized convex Orlicz function, the set $C_{M,1}\subseteq \mathcal {C}([0,1/2])$ is the smallest closed convex set containing $\{M_b :0<b\le 1\}$. Note that every function in $C_{M,1}$ extends to a normalized convex Orlicz function. So, we can safely define $\ell _F$ for $F\in C_{M,1}$.

Theorem 8

(cf. [15, Theorem 4.a.8]) Let M and F be normalized convex Orlicz functions. Assume that M is non-degenerate and that $F\in C_{M,1}$. Then, there is a block basic sequence of the unit vector system of $\ell _M$ that is equivalent to the unit vector system of $\ell _F$.

Proof

Lindenstrauss–Tzafriri’s proof of the “if” part of [15, Theorem 4.a.8] gives exactly this result. $\square$

Given $p\in [1,\infty )$, $F_p$ will denote the potential function given by $F_p(t)=t^p$, $t\ge 0$. We denote by $F_\infty$ the degenerate Orlicz function defined by $M(t)=0$ if $0\le t \le 1/2$ and $M(t)=2t-1$ if $t>1/2$. Of course, $\ell _{F_p}=h_{F_p}=\ell _p$ for $1\le p<\infty$, $\ell _{F_\infty }=\ell _\infty$, and $h_{F_\infty }=c_0$.

Theorem 9

(see [15, Comments below Theorem 4.a.9]) Let M be a non-degenerate normalized convex Orlicz function and let $p\in [\alpha _M,\beta _M]$. Then $F_p\in C_{M,1}$.

Proof

Lindenstrauss–Tzafriri’s proof of the “if” part of [15, Theorem 4.a.9] contains a proof of this result. $\square$

We say that a function $M:[0,\infty )\rightarrow [0,\infty )$ satisfies the $\varDelta _2$-condition at zero if there is $a>0$ such that $M(2t)\lesssim M(t)$ for $0\le t \le a$. Note that a non-degenerate normalized convex Orlicz function M satisfies the $\varDelta _2$-condition at zero if and only if $M(t)\lesssim M(t/2)$ for $0\le t \le 1$.

Theorem 10

(cf. [15, Proof of Theorem 4.a.9]) Let M be a non-degenerate normalized convex Orlicz function. The following are equivalent.

• (i) $\beta _M<\infty$.

• (ii) M satisfies the $\varDelta _2$-condition at zero.

• (iii) $\ell _M=h_M$.

• (iv) $F_\infty \notin C_{M,1}$.

Proof

First, we prove (i) $\Longrightarrow$ (ii). Assume that $\beta _M<\infty$. Then there are $1\le q ,C<\infty$ such that $M(b)\le C t^{-q} M(bt)$ for every $(b,t)\in (0,1]^2$. In particular, $M(b)\le C 2^q M(b/2)$ for every $0<b\le 1$.

(ii) $\Longrightarrow$ (iii) is a part of [15, Proposition 4.a.4].

Let us we prove (iii) $\Longrightarrow$ (iv). If $\ell _M=h_M$ then, by [15, Proposition 4.a.4], the unit vector system is a boundedly complete basis of $\ell _M$. Then, by [4, Theorem 3.3.2], no basic sequence of the unit vector system of $\ell _M$ is equivalent to the unit vector system of $\ell _\infty =\ell _{F_\infty }$. Therefore, by Theorem 8, $F_\infty \notin C_{M,1}$.

We go on by proving (iv) $\Longrightarrow$ (ii). Assume that $F_\infty \notin C_{M,1}$. Then there is constant $c>0$ such that

$$\begin{aligned} \sup \{ M_b(t) :0\le t \le 1/2\}=M_b(1/2)\ge c \end{aligned}$$

for every $b\in (0,1]$. In other words, $M(b)\le c^{-1} M(b/2)$ for every $0\le b \le 1$.

Finally, we prove (ii) $\Longrightarrow$ (i). Let $C\ge 2$ be such that $M(b)\le C M(b/2)$ for every $b\in (0,1]$. Choose $q=\log _2(C)$. Given $0<t\le 1$, pick $n\in \mathbb {N}$ such that $2^{-n}<t\le 2^{-n+1}$. We have

$$\begin{aligned} M(b)\le C^n M(2^{-n} b)=C 2^{(n-1)q} M(2^{-n} b)\le C t^{-q} M(tb). \end{aligned}$$

Therefore,

$$\begin{aligned} \inf \{ t^{-q} M_b(t) :0<b\le 1\}\ge C^{-1}>0. \end{aligned}$$

Consequently, $\beta _M\le q <\infty$. $\square$

For tackling the proof of Theorem 7 we need to study functions constructed from sequences belonging to Orlicz spaces. Given a normalized convex Orlicz function M and $f=(b_j)_{j=1}^\infty \in \mathbb {F}^\mathbb {N}$ we define

$$\begin{aligned} M_f:[0,\infty )\rightarrow [0,\infty ], \quad s\mapsto \sum _{j=1}^\infty M(|b_j| s). \end{aligned}$$

Lemma 6

Let M be a normalized convex Orlicz function and let $f\in \mathbb {F}^\mathbb {N}$ with $0<R:=\Vert f \Vert _{\ell _M}<\infty$. We have the following.

• (i) $\{ s \in [0,\infty ] :M_f(s)\le 1\}=[0,1/R]$.

• (ii) If there is $s>1/R$ such that $M_f(s)<\infty$, then $M_f(1/R)=1$.

• (iii) $M_f$ is convex in $[0,\infty ]$.

Proof

By definition, $M_f(s)\le 1$ if $s<1/R$, and $M_f(s)> 1$ if $s>1/R$. By the Monotone Convergence Theorem, $M_f(1/R)\le 1$. Consequently, Ad (i) holds. By the Dominated Convergence Theorem, $M_f$ is continuous on the interval $\{ s\in [0,\infty ) :M_f(s)<\infty \}$. Therefore, Ad (ii) also holds. The proof of Ad (iii) is straightforward. $\square$

Lemma 6 alerts us that, even if $f\in S_{\ell _M}$, $M_f(1)$ may be different from 1. This drawback, caused by dealing with $\ell _M$ instead of its separable part $h_M$, motivates the following definition.

Definition 2

Let M be a normalized Orlicz function and let $f\in S_{\ell _M}$. We define $N_f:[0,\infty )\rightarrow [0,\infty )$ by

$$\begin{aligned} N_f(t)={\left\{ \begin{array}{ll} M_f(t) &{}\quad \text { if } 0\le t \le 1/2, \\ (1-M_f(1/2))(2t-1)+M_f(1/2)&{}\quad \text { if } 1/2\le t <\infty . \end{array}\right. } \end{aligned}$$

Lemma 7

Let M be a normalized Orlicz function and let $f\in S_{\ell _M}$. Then $N_f$ is a normalized Orlicz function with $M_f(t)\le N_f(t)$ for every $0\le t \le 1$ and $M_f(t)=N_f(t)$ for every $0\le t \le 1/2$.

Proof

By definition, $N_f(1)=1$, and $N_f$ is linear in $[1/2,\infty )$. The function $M_f$ is convex in [0, 1], and $M_f(1)\le 1$. Consequently, for $1/2\le t \le 1$,

$$\begin{aligned} \frac{M_f(t)-M_f(1/2)}{t-1/2}\le \frac{M_f(1)-M_f(1/2)}{1-1/2} \le \frac{1-M_f(1/2)}{1-1/2} = \frac{N_f(t)-N_f(1/2)}{t-1/2} \end{aligned}$$

and, hence, $M_f(t)\le N_f(t)$. It is clear that $N_f(0)=0$, that $N_f$ is continuous, and that $N_f$ is convex in [0, 1 / 2]. If $0\le s \le 1/2 \le t$,

$$\begin{aligned} \frac{N_f(1/2)-N_f(s)}{1/2-s} \le \frac{M_f(1)-M_f(1/2)}{1-1/2} \le \frac{N_f(t)-N_f(1/2)}{t-1/2}. \end{aligned}$$

We infer that $N_f$ is convex. $\square$

Lemma 8

Let M be a normalized convex Orlicz function and $\mathcal {F}=(f_n)_{n=1}^\infty$ be a disjointly supported sequence in $\ell _M$. Then, the basic sequence $\mathcal {F}$ is isometrically equivalent to the unit vector basis of the sequence space defined from the modular

$$\begin{aligned} m_\mathcal {F}(f)=\sum _{n=1}^\infty M_{f_n}(|a_n|), \quad f =(a_n)_{n=1}^\infty \in \mathbb {F}^\mathbb {N}. \end{aligned}$$

(10)

Proof

Choose $\nu :\mathbb {N}\rightarrow \mathbb {N}$ such that $j\in {{\,\mathrm{supp}\,}}(f_{\nu (j)})$ and $j\notin {{\,\mathrm{supp}\,}}(f_n)$ if $n\not =\nu (j)$. Put $f_n=(b_{j,n})_{j=1}^\infty$ for every $n\in \mathbb {N}$. Given $f=(a_n)_{n=1}^\infty \in \mathbb {F}^\mathbb {N}$ we have $\sum _{n=1}^\infty a_n \, f_n = (a_{\nu (j)} b_{j,\nu (j)})_{j=1}^\infty$. Consequently,

$$\begin{aligned} m_M\left( \sum _{n=1}^\infty a_n \, f_n\right) =\sum _{j=1}^\infty M( |a_{\nu (j)} b_{j,\nu (j)}|) =\sum _{j=1}^\infty \sum _{n=1}^\infty M(|a_nb_{j,n}|) =m_\mathcal {F}(f). \end{aligned}$$

This equality between modulars yields the desired inequality between norms. $\square$

Lemma 9

Let M be a normalized convex Orlicz function and $\mathcal {F}=(f_n)_{n=1}^\infty$ be a disjointly supported sequence. Suppose that $\Vert f_n\Vert _{\ell _M}=1$ and that $\lambda =\inf _n M_{f_n}(1/2)>0$. Then, if $\mathbf {N}=(N_{f_n})_{n=1}^\infty$,

$$\begin{aligned} \left\| \sum _{n=1}^\infty a_n \, f_n\right\| _{\ell _M}\approx \left\| (a_n)_{n=1}^\infty \right\| _{\ell _\mathbf {N}} \end{aligned}$$

for $(a_n)_{n=1}^\infty \in \mathbb {F}^\mathbb {N}$.

Proof

Notice that $M_{f_n}(1/2)\le M_{f_n}(1) \le 1$ for every $n\in \mathbb {N}$ and, hence, $\lambda \le 1$. In light of Lemma 8, we need to prove that, if $\Vert \cdot \Vert _\mathcal {F}$ denotes the norm induced by $m_\mathcal {F}$, the norms $\Vert \cdot \Vert _{\ell _\mathbf {N}}$ and $\Vert \cdot \Vert _\mathcal {F}$ are equivalent.

By Lemma 7, $m_\mathcal {F}(f)\le m_\mathbf {N}(f)$ whenever $m_\mathbf {N}(f)\le 1$. Consequently, $\Vert f \Vert _\mathcal {F}\le \Vert f \Vert _{\ell _\mathbf {N}}$ for every $f\in \mathbb {F}^\mathbb {N}$.

Conversely, suppose that $\Vert f \Vert _\mathcal {F}<\lambda$. Then, $m_\mathcal {F}(f/\lambda )\le 1$. The convexity of $M_{f_n}$ yields, if $f=(a_n)_{n=1}^\infty$,

$$\begin{aligned} M_{f_n}(|a_n|)\le \lambda M_{f_n}\left( \frac{|a_n|}{\lambda }\right) \le \lambda \le M_{f_n}\left( \frac{1}{2}\right) , \quad n\in \mathbb {N}. \end{aligned}$$

Consequently, $|a_n|\le 1/2$ for every $n\in \mathbb {N}$. Therefore,

$$\begin{aligned} m_\mathbf {N}(f)=\sum _{n=1}^\infty M_{f_n} (|a_n|)\le \sum _{n=1}^\infty M_{f_n}\left( \frac{|a_n|}{\lambda }\right) \le 1. \end{aligned}$$

Hence, $\Vert f \Vert _{\ell _\mathbf {N}}\le 1$. We infer that $\Vert f \Vert _{\ell _\mathbf {N}}\le \lambda ^{-1} \Vert f \Vert _\mathcal {F}$ for every $f\in \mathbb {F}^\mathbb {N}$. $\square$

Proposition 6

Let M be a non-degenerate normalized convex Orlicz function. Then $M_f\in C_{M,1}$ for every $f\in S_{\ell _M}$.

Proof

Assume, without loss of generality, that $f=(b_j)_{j=1}^\infty \in [0,\infty )^\mathbb {N}$. Denote $\lambda _j=M(b_j)$ for $j\in B:={{\,\mathrm{supp}\,}}(f)$. By Part (i) of Lemma 6, $\sum _{j\in B} \lambda _j\le 1$. Denote $\lambda _\infty :=1-\sum _{j\in B} \lambda _j$. In the case when $\lambda _\infty =0$ we have

$$\begin{aligned} \sum _{j\in B} \lambda _j=1 \quad \text {and}\quad M_f=\sum _{j\in B} \lambda _j M_{b_j}. \end{aligned}$$

In the case when $\lambda _\infty >0$, by Part (ii) of Lemma 6 and the identity (8), $f\in \ell _M\setminus h_M$. Therefore, by Theorem 10, $F_\infty \in C_{M,1}$. The identities

$$\begin{aligned} \lambda _\infty + \sum _{j\in B} \lambda _j=1 \quad \text {and}\quad M_f(t)=\lambda _\infty F_\infty (t)+ \sum _{j\in B} \lambda _j M_{b_j}(t) \end{aligned}$$

for every $t\in [0,1/2]$ yield that, in both cases, $M_f$ is a (possibly infinite) convex combination of functions in $C_{M,1}$. Consequently, $M_f\in C_{M,1}$. $\square$

We are now in a position to complete the proof of the main theorem of this section.

Proof of Theorem 7

(i) $\Longrightarrow$ (iii) follows from combining Theorem 9 with Theorem 8, and (iii) $\Longrightarrow$ (ii) is obvious. In order to prove that (ii) implies (i), we pick a disjointly supported sequence $\mathcal {F}=(f_n)_{n=1}^\infty$ in $\ell _M$ equivalent to the unit vector basis of $\ell _p$. By unconditionality, we can assume, without loss of generality, that $\Vert f_n\Vert _{\ell _M}=1$ for every $n\in \mathbb {N}$. Then, by Proposition 6, $M_{f_n}\in C_{M,1}$ for every $n\in \mathbb {N}$.

We consider two opposite situations according to the behavior of the numbers $\lambda _n=M_{f_n}(1/2)\in [0,1]$ for $n\in \mathbb {N}$.

Assume that $\inf _n \lambda _n=0$. Then, passing to a suitable subsequence, we can suppose that $\sum _{n=1}^\infty \lambda _n \le 1$. If $m_\mathcal {F}$ is as in (10) and $\Vert \cdot \Vert _\mathcal {F}$ is its associated norm, we have

$$\begin{aligned} m_\mathcal {F}\left( \frac{ f}{2\Vert f \Vert _\infty }\right) \le 1 \end{aligned}$$

and, hence, $\Vert f\Vert _\mathcal {F}\le 2\Vert f\Vert _\infty$ for every $f\in \mathbb {F}^\mathbb {N}$. Consequently, by Lemma 8, the unit vector system of $\ell _\infty$ dominates the basic sequence $\mathcal {F}$. We infer that $p=\infty$. Moreover $\lim _n M_{f_n}=0$ uniformly in [0, 1 / 2] and, hence, applying Proposition 6, we obtain $F_\infty \in C_{M,1}$.

Assume that $\inf _n \lambda _n>0$. Taking into account that $C_{M,1}$ is compact by [15, Lemma 4.a.6], passing to a suitable subsequence, we can suppose that there is $F\in C_{M,1}$ such that

$$\begin{aligned} \sup _{0\le t \le 1/2} |M_{f_n}(t) -F(t)|\le 2^{-n} \end{aligned}$$

(11)

for every $n\in \mathbb {N}$. Therefore, if $\mathbf {N}=(N_{f_n})_{n=1}^\infty$, applying Theorem 6 yields $\ell _\mathbf {N}=\ell _F$. Consequently, by Lemma 9, $\ell _p=\ell _F$.

In both cases, there is $F\in C_{M,1}$ such that $\ell _F=\ell _p$. By Remark 2, for such a function F, there is $a>0$ such that $F(t)\approx F_p(t)=t^p$ for $0\le t\le a$.

Let $r<\alpha _M$. There is a constant $C_1<\infty$ such that $M_b(t) \le C_1 t^r$ for every $0<b\le 1$ and every $0\le t \le 1$. By convexity and continuity, $N(t) \le C_1 t^r$ for every $N\in C_{M,1}$ and every $0\le t \le 1/2$. Consequently, there is $C_2<\infty$ such that $F_p(t)\le C_2 t^r$ for every $0\le t\le a$. We infer that $r\le p$. Letting r tend to $\alpha _M$ we obtain $\alpha _M\le p$. We prove that $p\le \beta _M$ in a similar way. $\square$

Theorem 11

Let M be a non-degenerate normalized convex Orlicz function. Then the unit vector system of $\ell _M$ is non-$\ell _1$-splicing if and only if $1<\alpha _M$.

Proof

Taking into account the identity (9), the result follows from combining Theorem 4 with Theorem 7. $\square$

Remark 3

Define, for an Orlicz function M,

$$\begin{aligned} \varOmega _n=\inf \left\{ \rho >0 :\sup _{0<t\le 1} \frac{ M(t/\rho )}{M(t)}\le \frac{1}{n}\right\} , \quad n\in \mathbb {N}. \end{aligned}$$

Lechner [12, Theorem 6.3] proved that if $\lim _n \varOmega _n/n=0$ then the unit vector system of $\ell _M$ is non-$\ell _1$-splicing. Let us give a proof based on Theorem 11 for this result. Assume that $\lim _n \varOmega _n/n=0$. In particular, there is $n\ge 3$ such that $\varOmega _n/n<1/2$. Consequently, there is $\rho \le R:=n/2$ with

$$\begin{aligned} \frac{ M(t/\rho )}{M(t)}\le \frac{1}{n}, \quad 0<t\le 1. \end{aligned}$$

Therefore, $M(R^{-1}t)\le (2R)^{-1}M(t)$ for every $t\in (0,1]$. We deduce by induction that $M(R^{-k+1}t)\le (2R)^{-k+1}M(t)$ for every $t\in (0,1]$ and $k\in \mathbb {N}$. Since $R>1$, $q:=1+\log _R 2>1$. Let $u\in (0,1]$ and pick $k\in \mathbb {N}$ with $R^{-k}<u\le R^{-k+1}$. We have

$$\begin{aligned} M(ut)\le M(R^{-k+1}t)\le (2R)^{-k+1} M(t)=2R R^{-qk}M(t) \le 2R u^q M(t). \end{aligned}$$

Consequently, $q\in A_M$ and, hence, $\alpha _M>1$.

As in Sect. 3, we close by telling apart Orlicz sequence spaces from $\ell _\infty$ and writing down the straightforward consequence of combining Theorem 1 with Theorem 11. We emphasize that, in light of Theorem 2 and the results achieved in [9], Theorem 12 is a novelty only in the case when $\ell _M$ is non-separable, i.e., when $\beta _M=\infty$.

Proposition 7

Let M be a normalized convex Orlicz function. Then $\ell _M$ is a $\mathcal {L}_\infty$-space if and only if M is degenerate.

Proof

Let $M^*$ be the complementary Orlicz function of M. If $\ell _M$ were a $\mathcal {L}_\infty$-space, then $h_{M^*}$ would be a $\mathcal {L}_1$-space. Since the unit vector basis of $h_{M^*}$ is unconditional, we would obtain $h_{M^*}=\ell _1$. Therefore, $\ell _M=\ell _\infty =\ell _{F_\infty }$. Consequently, there would be $0<a\le 1/2$ such that $M(t)=F_\infty (t)=0$ for every $0\le t \le a$. $\square$

Theorem 12

Let $p\in [1,\infty ]$ and M be a non-degenerate normalized convex Orlicz function with $\alpha _M>1$. Let $\mathbb {Y}=\ell _M$ or $\mathbb {Y}=\ell _p(\ell _M)$. Then, if $T\in \mathcal {L}(\mathbb {Y})$, the identity map on $\mathbb {Y}$ factors through either T or $\mathrm {Id}_\mathbb {Y}-T$. Consequently, $\ell _p(\ell _M)$ is a primary Banach space.