1 Introduction and preliminaries
Let \(c_{00}\) be the space of all scalar sequences, that is, elements of \(\mathbb{R}^{\mathbb{N}}\) or \(\mathbb{C}^{N}\) with finitely many nonzero entries. Let X and Y be infinite-dimensional Banach spaces. If \((x_{k})_{k=1}^{\infty}\) is a basis of the Banach space X and \(x= \sum_{k=1}^{\infty}a_{k}x_{n}\), then we denote \(\operatorname{supp}x= \{k\in\mathbb{N}:a_{k}\neq0\}\), that is, the set of indices corresponding to the nonzero entries of x, which is called the support of x. When the basis \((x_{k})_{k=1}^{\infty}\) is clear from context, we write \(x=(a_{k})_{k=1}^{\infty}\) instead of \(x= \sum_{k=1}^{\infty}a_{k}x_{n}\).
Now, suppose that
\((x_{k})_{k=1}^{\infty}\) and
\((y_{k})_{k=1}^{\infty }\) are sequences in
X and
Y, respectively. If there exists
\(C\in [1, \infty)\) such that
$$\Biggl\Vert \sum_{k=1}^{\infty}a_{k}x_{k} \Biggr\Vert _{X}\leq C \Biggl\Vert \sum _{k=1}^{\infty}a_{k}y_{k} \Biggr\Vert _{Y} $$
for all finitely supported sequences
\((a_{k})_{k=1}^{\infty}\in c_{00}\), then
\((y_{k})_{k=1}^{\infty}\)
C-dominates
\((x_{k})_{k=1}^{\infty}\), and we can write
\((x_{k})_{k=1}^{\infty }\lesssim_{C}(y_{k})_{k=1}^{\infty}\). In case
C does not matter, we will simply say that
\((y_{k})_{k=1}^{\infty}\) dominates
\((x_{k})_{k=1}^{\infty}\) and
\((x_{k})_{k=1}^{\infty}\lesssim (y_{k})_{k=1}^{\infty}\). If
\((x_{k})_{k=1}^{\infty}\lesssim _{C}(y_{k})_{k=1}^{\infty}\) and
\((y_{k})_{k=1}^{\infty}\lesssim _{C}(x_{k})_{k=1}^{\infty}\), then
\((x_{k})_{k=1}^{\infty}\) and
\((y_{k})_{k=1}^{\infty}\) are
C-equivalent, and we can write it as
\((x_{k})_{k=1}^{\infty}\approx_{C}(y_{k})_{k=1}^{\infty}\).
A basic sequence
\((x_{k})_{k=1}^{\infty}\) is called subsymmetric just in case it is unconditional and equivalent to each of its subsequences. It is called symmetric whenever it is unconditional and equivalent to each of its permutations. Lindenstrauss and Tzafriri [
1] studied the uniform boundedness which means that if
\((x_{k})_{k=1}^{\infty}\) is a subsymmetric basic sequence, then there is a uniform constant
\(C \geq1\) such that if
\((x_{k_{n}})_{n=1}^{\infty}\) is any subsequence and
\((\varepsilon _{n})_{n=1}^{\infty}\) is any sequence of signs, then
\((x_{k})_{k=1}^{\infty}\) is
C-equivalent to
\((\varepsilon _{n}x_{k_{n}})_{n=1}^{\infty}\). In this case, we say that
\((x_{k})_{k=1}^{\infty}\) is
C-subsymmetric. Similarly, if
\((x_{k})_{k=1}^{\infty}\) is symmetric, then there is
\(C \geq1\) such that
\((x_{k})_{k=1}^{\infty}\) is
C-equivalent to each
\((\varepsilon _{k}x_{\varrho(k)})_{k=1}^{\infty}\), where
ϱ is a permutation of
\(\mathbb{N}\), and in this case, we say that
\((x_{k})_{k=1}^{\infty}\) is
C-symmetric. Note that
C-symmetry implies
C-subsymmetry, which in turn implies
C-unconditionality. In this paper, we use standard facts and notation from Banach spaces and approximation theory. For necessary background, see [
1,
2], and references therein. We denote by
\(\mathbb{F}\) the real or complex field. The canonical basis of
\(\mathbb{F}\) is denoted by
\((e_{k})_{k=1}^{\infty}\), that is,
\(e_{k}=(\delta _{k,n})_{n=1}^{\infty}\), where
\(\delta_{k,n}=1\) if
\(n=k\) and
\(\delta _{k,n}=0\) if
\(n\neq k\).
An Orlicz function M is a function that is continuous, nondecreasing, and convex with \(M(0) = 0\), \(M(x)>0\) for \(x>0\) and \(M(x) \longrightarrow\infty\) as \(x \longrightarrow\infty\). An Orlicz function M is said to satisfy \(\Delta_{2}\)-condition if there exists \(R > 0\) such that \(M(2u) \leq R M(u)\), \(u \geq0\).
The study of Orlicz sequence spaces was initiated with a certain specific purpose in Banach space theory. Lindenstrauss and Tzafriri [
3] used the idea of an Orlicz function to define the sequence space
$$\ell_{M} = \Biggl\{ x \in w : \sum^{\infty}_{k=1} M \biggl(\frac{ \vert x_{k} \vert }{\rho}\biggr) < \infty \text{ for some } \rho>0\Biggr\} , $$
which is called an Orlicz sequence space. The space
\(\ell_{M}\) is a Banach space with the norm
$$\Vert x \Vert = \inf\Biggl\{ \rho> 0 : \sum^{\infty}_{k=1} M\biggl(\frac{ \vert x_{k} \vert }{\rho}\biggr) \leq 1\Biggr\} . $$
It is shown in [
3] that every Orlicz sequence space
\(\ell_{M}\) contains a subspace isomorphic to
\(\ell_{p}\) (
\(p \geq1\)).
A sequence
\(\mathcal{M} = (M_{k})\) of Orlicz functions is called a
Musielak-Orlicz function (see [
4,
5]). For more detail about sequence spaces, see [
6‐
26], and references therein.
The notion of a difference operator in the sequence spaces was first introduced by Kızmaz [
27]. The idea of difference sequence spaces of Kızmaz was further generalized by Et and Çolak [
28]. Later, this concept was studied by Bektaş et al. [
29] and Et et al. [
30]. Now, the difference matrix
\(\Delta= \delta_{nk}\) defined by
$$\delta_{nk}= \textstyle\begin{cases} (-1)^{n-k} & (n-1\leq k\leq n), \\ 0 & (0\leq k< n-1 \text{ or } n> k). \end{cases} $$
The difference operator of order
m is defined by
\(\Delta^{m}: w\rightarrow w\),
\((\Delta^{1}x)_{k}= (x_{k}-x_{k-1})\) and
\(\Delta ^{m}x= (\Delta^{1}x)_{k}\circ(\Delta^{m-1}x)_{k}\) for
\(m\geq2\).
The triangle matrix
\(\Delta^{(m)}=\delta^{(m)}_{nk}\) defined by
$$\delta^{(m)}_{nk}= \textstyle\begin{cases} (-1)^{n-k}\big( {\scriptsize\begin{matrix}{} m \cr n-k \end{matrix}} \big) & (\max\{0,n-m\}\leq k\leq n), \\ 0 & (0\leq k< \max\{0, n-m\} \text{ or } n> k) \end{cases} $$
for all
\(k,n \in\mathbb{N}\) and any fixed
\(m\in\mathbb{N}\).
The Lorentz sequence space was introduced by Lorentz [
31,
32]. This space plays an important role in the theory of Banach spaces, whereas Garling [
33] studied, for
\(v=(k^{-1/2})_{k=1}^{\infty}\), the canonical unit vectors of
\(g(v, 1)\) from a subsymmetric basic sequence that is not symmetric. Pujara [
34] studied
\(g(v, 2)\) for
\(v=(k^{-1/2})_{k=1}^{\infty}\) and showed that these spaces are uniformly convex and that their canonical basis is subsymmetric but not symmetric. The space from [
35] can be viewed as being constructed from variations of
\(g(w, 1)\) for various choices of weights and gives a canonical basis that is 1-greedy and subsymmetric but not symmetric. Both the Garling sequence spaces and the Lorentz sequence spaces are defined by taking the completion of
\(c_{00}\) under the norms
\(\Vert\cdot\Vert_{g}\) or
\(\Vert\cdot \Vert_{d}\), respectively. The only difference between these norms is that
\(\Vert\cdot\Vert_{g}\) is defined by taking a certain supremum over subsequences instead of permutations of sequences as is the case for
\(\Vert\cdot\Vert_{d}\). Wallis [
36] generalized the construction of Garling for each
\(1\leq p< \infty\) and normalized nonincreasing sequence of positive numbers that exhibited complementably homogeneous Banach Garling space related to the Lorentz sequence space. For more detail about these spaces, see [
37‐
39], and references therein.
Let us consider the set of weights
$$V= \bigl\{ (v_{k})_{k=1}^{\infty}\in c_{0} \setminus l_{1}:1=v_{1}\geq v_{2}>\cdots\geq v_{k}\geq v_{k+1}\geq\dots>0\bigr\} . $$
Let
\(\mathcal{M}=(M_{k})\) be a sequence of Orlicz functions, and let
\(v=(v_{k})_{k=1}^{\infty}\in V\) be a weight, that is, a sequence of positive scalars. Then the Orlicz-Garling sequence space is defined as the Banach space consisting of all scalar sequences
\(A= (a_{k})_{k=1}^{\infty}\) such that
$$\begin{aligned} &\Vert A \Vert _{g(\mathcal{M},\Delta^{(m)},v,p)}\\ &\quad = \inf\Biggl\{ \rho>0: \sup _{\psi\in\mathcal{O}}\Biggl(\sum_{k=1}^{\infty} \biggl(M_{k}\biggl(\frac{ \vert \Delta^{(m)}a_{\psi(k)} \vert ^{p}v_{k}}{\rho}\biggr)\biggr)\Biggr)^{\frac{1}{p}} \leq1 \text{ for some } \rho>0\Biggr\} , \end{aligned} $$
where
\(\mathcal{O}\) denotes the set of all increasing functions from
\(\mathbb{N}\) to
\(\mathbb{N}\). We will assume that
v is normalized, that is,
\(v_{1}=1\). We consider the normed space
\(g(\mathcal{M},\Delta ^{(m)},v, p)= \{A= (a_{k})_{k=1}^{\infty}: \Vert A \Vert_{g(\mathcal {M},\Delta^{(m)},v,p,)}<\infty\}\).
Let Π denote the set of permutations on
\(\mathbb{N}\), and let
\(\mathcal{M}=(M_{k})\) be a sequence of Orlicz functions. Then, for any
\(1\leq p< \infty\) and
\(v\in V\), we define the Orlicz-Lorentz sequence space
\(d(\mathcal{M},\Delta^{(m)},v,p)\) as the completion of
\(c_{00}\) under the norm
\(\Vert\cdot\Vert_{d(\mathcal{M},\Delta ^{(m)},v,p)}\) defined by
$$\begin{aligned} &\Vert A \Vert _{d(\mathcal{M},\Delta^{(m)},v,p)}\\ &\quad = \inf\Biggl\{ \rho>0: \sup _{\varrho\in\Pi}\Biggl(\sum_{k=1}^{\infty } \biggl(M_{k}\biggl(\frac{ \vert \Delta^{(m)}a_{\varrho(k)} \vert ^{p}v_{k}}{\rho}\biggr)\biggr)\Biggr)^{\frac{1}{p}} \leq1 \text{ for some } \rho>0\Biggr\} . \end{aligned} $$
We consider the normed space
\(d(\mathcal{M},\Delta^{(m)},v, p)= \{A= (a_{k})_{k=1}^{\infty}: \Vert A \Vert_{d(\mathcal{M},\Delta ^{(m)},v,p,)}<\infty\}\).
The main aim of this paper is to introduce and study some difference Orlicz-Garling sequence spaces and Orlicz-Lorentz sequence spaces. Using the originally developed Orlicz-Lorentz sequence space, we show that the Orlicz-Garling sequence space admits a unique 1-subsymmetric basis. Finally, we discuss some additional geometric properties of \(g(\mathcal{M},\Delta^{(m)}, v, p)\) and also establish some inclusion relations between these spaces.
2 Main results on \(g(\mathcal{M},\Delta ^{(m)}, v, p)\)
Given a function
ψ, we denote by
\(U(\psi)\) its range. Let
\(\mathcal{M}=(M_{k})\) be a sequence of Orlicz functions, and let
\(v=(v_{k})_{k=1}^{\infty}\in V\) be a weight. Let
\(\mathcal{O}_{A}\) be the set of increasing functions from an integer interval
\([1,\ldots,u]\cap \mathbb{N}\) into
\(\mathbb{N}\). For a given
\(\psi\in\mathcal {O}_{A}\), we denote by
\(u(\psi)\) the largest integer in its domain; a function in
\(\mathcal{O}_{A}\) is univocally determined by its range. For
\(A=(a_{k})_{k=1}^{\infty}\), we have
$$ \begin{aligned}[b] \Vert A \Vert _{g}^{\mathcal{M},\Delta^{(m)},p}&= \inf\Biggl\{ \rho>0: \sup_{\psi\in\mathcal{O}_{A}}\sum _{k=1}^{u(\psi)}\biggl(M_{k}\biggl( \frac{ \vert \Delta^{(m)}a_{\psi(k)} \vert ^{p}v_{k}}{\rho}\biggr)\biggr)\leq1 \\ &\quad \text{ for some } \rho>0\Biggr\} . \end{aligned} $$
(2.1)
A Banach space with an unconditional basis is said to have a unique unconditional if any two seminormalized unconditional bases of
X are equivalent.
Now we define the weak Orlicz-Lorentz sequence space
\(d_{\infty }(\mathcal{M},\Delta^{(m)}, v, p)\) for a sequence of Orlicz functions
\(\mathcal{M}=(M_{k})\). For
\(v\in V\) and
\(1\leq p< \infty\), it consists of all sequences
\(A=(a_{k})_{k=1}^{\infty}\in c_{0}\) such that
$$\Vert A \Vert _{d_{\infty}(\mathcal{M},\Delta ^{(m)},v,p)}= \inf\Biggl\{ \rho>0: \sup_{k} \Biggl(\sum_{n=1}^{k}\biggl(M_{k} \biggl(\frac{ \vert \Delta^{(m)}a_{k}^{*} \vert v_{n}}{\rho}\biggr)\biggr)\Biggr)^{\frac{1}{p}}\leq1 \text{ for some } \rho>0\Biggr\} , $$
where
\((a_{k}^{*})_{k=1}^{\infty}\) denotes the decreasing rearrangement of
A.
Let
\((x_{k})_{k=1}^{\infty}\) be a basis for a Banach space
X, and let
\(D= \sum_{k=1}^{\infty}b_{k}x_{k}\) be a vector in
X. For any
\(A= \sum_{k=1}^{\infty}a_{k}x_{k}\in X\), it can be written as
\(A\prec D\) if
(i)
\(\operatorname{supp}(A)\subseteq \operatorname{supp}(D)\),
(ii)
\(a_{k}=b_{k}\) for all \(k\in \operatorname{supp}(A)\), and
(iii)
\(\Vert A \Vert= \Vert D \Vert\).
We can say that D is minimal in X if it is minimal in X equipped with the partial order ≺, that is, if \(A\prec D\) implies \(A=D\). If D is supported with respect to a basis \((x_{k})_{k=1}^{\infty}\), then there exists a minimal \(A\prec D\).
A block basic sequence of a basic sequence
\((x_{k})_{k=1}^{\infty}\) in a Banach space
X is a sequence
\((y_{k})_{k=1}^{\infty}\) of nonzero vectors of the form
$$y_{k}= \sum_{j=p_{k}}^{p_{k+1}-1}a_{j}x_{j} $$
for some increasing sequence of integers
\((p_{k})_{k=1}^{\infty}\) with
\(p_{1}=1\) and some
\((a_{k})_{k=1}^{\infty}\in\mathbb{F}^{\mathbb{N}}\).
Let us define some linear functions as follows:
(i)
For a given sequence of signs \(\varepsilon= (\varepsilon_{k})_{k=1}^{\infty}\), we define the linear mapping \(T_{\varepsilon}:\mathbb{F}^{\mathbb{N}}\rightarrow\mathbb {F}^{\mathbb{N}}\) by \(T_{\varepsilon}((a_{k})_{k=1}^{\infty})= (\varepsilon_{k}a_{k})_{k=1}^{\infty}\).
(ii)
The coordinate projection on
\(B\subseteq\mathbb{N}\) is defined by
\(P_{B}: \mathbb{F}^{\mathbb{N}}\rightarrow\mathbb {F}^{\mathbb{N}}\),
\(P_{B}((a_{k})_{k=1}^{\infty})= (\mu _{k}a_{k})_{k=1}^{\infty}\), where
$$\mu_{k}= \textstyle\begin{cases} 1 & \text{if } k\in B, \\ 0 & \text{if } n\notin B. \end{cases} $$
(iii)
Given \(\psi\in\mathcal{O}\), we define the linear mapping \(S_{\psi}: \mathbb{F}^{\mathbb{N}}\rightarrow\mathbb {F}^{\mathbb{N}}\) by \(S_{\psi}((a_{k})_{k=1}^{\infty})= (a_{\psi (k)})_{k=1}^{\infty}\), and
(iv)
we define the linear function
\(R_{\psi}: \mathbb {F}^{\mathbb{N}}\rightarrow\mathbb{F}^{\mathbb{N}}\) by
\(R_{\psi }((a_{k})_{k=1}^{\infty})= (b_{k})_{k=1}^{\infty}\), where
$$b_{k}= \textstyle\begin{cases} a_{n} & \text{if } k=\psi(n), \\ 0 & \text{if } k\notin\psi(n). \end{cases} $$
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