2.1 Noncommutative symmetric spaces

Throughout, $\mathcal {M}$ denotes a semifinite von Neumann algebra equipped with a faithful normal semifinite trace $\tau $ . Let $\widetilde {\mathcal {M}}$ denote the associated topological $*$ -algebra of $\tau $ -measurable operators in the sense of [Reference Fack and Kosaki12]. For $x \in \widetilde {\mathcal {M}}$ , we recall that its generalized singular number $\mu (x)$ is the real-valued function defined by

$$ \begin{align*} \mu_t(x)= \inf\big\{ s>0 : \tau\big(\chi_{(s,\infty)}(|x|)\big)\leq t\big\}, \quad t>0, \end{align*} $$

where $\chi _{(s,\infty )}(|x|)$ is the spectral projection of $|x|$ associated with the interval $(s,\infty )$ . We observe that if $\mathcal {M}$ is the abelian von Neumann algebra $L_\infty (0,\infty )$ with the trace given by integration with respect to the Lebesgue measure, then $\widetilde {\mathcal {M}}$ becomes the space of measurable complex functions on $(0,\infty )$ which are bounded except on a set of finite measure and for $f\in \widetilde {\mathcal {M}}$ , $\mu (f)$ is precisely the usual decreasing rearrangement of the function $|f|$ . We refer to [Reference Pisier and Xu38] for more information on noncommutative integration.

We denote by $L_0$ , the space of measurable functions on the interval $(0,\infty )$ . Recall that a quasi-Banach function space $(E,\|\cdot \|_E)$ of measurable functions on the interval $(0,\infty )$ is called symmetric if for any $g \in E$ and any $f \in L_0$ with $\mu (f) \leq \mu (g)$ , we have $f \in E$ and $\|f\|_E \leq \|g\|_E$ . Throughout, all function spaces are assumed to be defined on the interval $(0,\infty )$ .

Let E be a symmetric quasi-Banach function space. We define the corresponding noncommutative space by setting:

$$ \begin{align*} E(\mathcal{M}, \tau) = \big\{ x \in \widetilde{\mathcal{M}}\ : \ \mu(x) \in E \big\}. \end{align*} $$

Equipped with the quasi-norm $\|x\|_{E(\mathcal {M},\tau )} := \| \mu (x)\|_E$ , the linear space $E(\mathcal {M},\tau )$ becomes a complex quasi-Banach space [Reference Kalton and Sukochev28, Reference Xu50] and is usually referred to as the noncommutative symmetric space associated with $(\mathcal {M},\tau )$ corresponding to E. We remark that if $0< p<\infty $ and $E=L_p$ , then $E(\mathcal {M}, \tau )$ is exactly the usual noncommutative $L_p$ -space $L_p(\mathcal {M},\tau )$ associated with $(\mathcal {M},\tau )$ . In the sequel, $E(\mathcal {M},\tau )$ will be abbreviated to $E(\mathcal {M})$ .

Other classes of examples that are relevant for our purpose are the class of Orlicz spaces and the class of Lorentz spaces. We review these two classes for convenience.

A function $\Phi : [0,\infty ) \to [0,\infty )$ is called an Orlicz function whenever it is strictly increasing, continuous, $\Phi (0)=0$ , and $\lim _{u\to \infty }\Phi (u)=\infty $ . The Orlicz space $L_\Phi $ is the collection of all $f\in L_0$ for which there exists a constant c such that $I_\Phi (|f|/c)<\infty $ where the modular functional $I_\Phi (\cdot )$ is defined by:

$$ \begin{align*} I_\Phi(|g|)=\int_0^\infty \Phi(|g(t)|)\ dt, \quad g \in L_0. \end{align*} $$

The space $L_\Phi $ is equipped with the Luxemburg quasi-norm:

$$ \begin{align*} \big\|f \big\|_{L_\Phi}= \inf\big\{ c>0 : I_\Phi(|f|/c) \leq 1\big\}. \end{align*} $$

If $\Phi $ is convex then $L_\Phi $ is a symmetric Banach function space. However, we do not restrict ourselves to just the case of normed spaces. We refer to [Reference Turpin45] for Orlicz spaces associated with non necessarily convex functions.

We recall that for $0<p \leq q<\infty $ , $\Phi $ is called p-convex (resp., q-concave) if the function $t\mapsto \Phi (t^{1/p})$ (resp., $t\mapsto \Phi (t^{1/q})$ ) is convex (resp., concave). Below, we only consider Orlicz spaces associated with Orlicz functions that are p-convex and q-concave for some $0<p\leq q<2$ .

For the next relevant example, assume that $0<p,q\leq \infty $ . The Lorentz space $L_{p,q}$ is the space of all $f \in L_0$ for which $\|f\|_{p,q}<\infty $ where

$$ \begin{align*} \big\|f \big\|_{p,q} =\begin{cases} \left(\displaystyle{\int_{0}^\infty \mu_{t}^{q}(f)\ d(t^{q/p})}\right)^{1/q}, &0< q < \infty; \\ \displaystyle{\mathop{\mathrm{sup}}_{t>0} t^{1/p} \mu_t(f)}, &q= \infty. \end{cases} \end{align*} $$

If $1\leq q\leq p <\infty $ or $p=q=\infty $ , then $L_{p,q}$ is a symmetric Banach function space. If $1<p<\infty $ and $p\leq q\leq \infty $ , then $L_{p,q}$ can be equivalently renormed to become a symmetric Banach function [Reference Bennett and Sharpley2, Theorem 4.6]. In general, $L_{p,q}$ is only a symmetric quasi-Banach function space.

Both noncommutative Orlicz spaces and noncommutative Lorentz spaces will be heavily involved throughout the paper.

2.2 Martingale Hardy spaces and conditioned spaces

In the sequel, we always denote by $(\mathcal {M}_n)_{n \geq 1}$ an increasing sequence of von Neumann subalgebras of ${\mathcal {M}}$ whose union is w*-dense in $\mathcal {M}$ . For every $n\geq 1$ , we assume that there is a trace preserving conditional expectation $\mathcal {E}_n$ from ${\mathcal {M}}$ onto ${\mathcal {M}}_n$ .

Let E be a symmetric quasi-Banach function space and $x=(x_n)_{n\geq 1}$ be a martingale. If for every $n\geq 1$ , $x_n \in E(\mathcal {M}_n)$ , then we say that $(x_n)_{n\geq 1}$ is an $E(\mathcal {M})$ -martingale. In this case, we set

$$ \begin{align*}\| x \|_{E(\mathcal{M})}= \mathop{\mathrm{sup}}_{n \geq 1} \| x_n \|_{E(\mathcal{M})}. \end{align*} $$

If $\| x \|_{E(\mathcal {M})} < \infty $ , then x will be called a bounded $E(\mathcal {M})$ -martingale.

For a martingale $x=(x_n)_{n\geq 1}$ , we set $dx_n=x_n-x_{n-1}$ for $n\geq 1$ with the usual convention that $x_0=0$ . The sequence $dx=(dx_n)_{n\geq 1}$ is called the martingale difference sequence of x. A martingale x is called a finite martingale if there exists N such that $dx_n=0$ for all $n\geq N$ .

Let us now review some basic definitions related to martingale Hardy spaces associated to noncommutative symmetric spaces.

Following [Reference Pisier and Xu37], we define the column square functions of a given martingale $x = (x_k)$ by setting:

$$ \begin{align*} S_{c,n} (x) = \left ( \sum^n_{k = 1} |dx_k |^2 \right )^{1/2}, \quad S_c (x) = \left ( \sum^{\infty}_{k = 1} |dx_k |^2 \right )^{1/2}. \end{align*} $$

The conditioned versions were introduced in [Reference Junge and Xu26]. For a given $L_2(\mathcal {M}) +\mathcal {M}$ -martingale $(x_k)_{k\geq 1}$ , we set

$$ \begin{align*} s_{c,n} (x) = \left ( \sum^n_{k = 1} \mathcal{E}_{k-1}|dx_k |^2 \right )^{1/2}, \quad s_c (x) = \left ( \sum^{\infty}_{k = 1} \mathcal{E}_{k-1}|dx_k |^2 \right )^{1/2}. \end{align*} $$

The operator $s_c(x)$ is called the column conditioned square function of x. For convenience, we will use the notation

$$ \begin{align*} \mathcal{S}_{c,n}(a)= \left ( \sum^n_{k = 1} |a_k |^2 \right )^{1/2}, \quad \mathcal{S}_c (a) = \left ( \sum^{\infty}_{k = 1} |a_k |^2 \right )^{1/2}\, \end{align*} $$

and

$$ \begin{align*} \sigma_{c,n} (b) = \left ( \sum^{n}_{k = 1} \mathcal{E}_{k-1}|b_k |^2 \right )^{1/2}, \quad \sigma_c (b) = \left ( \sum^{\infty}_{k = 1} \mathcal{E}_{k-1}|b_k |^2 \right )^{1/2} \end{align*} $$

for sequences $a=(a_k)_{k\geq 1}$ in $L_1(\mathcal {M})+\mathcal {M}$ and $b=(b_k)_{k\geq 1}$ in $L_2(\mathcal {M}) +\mathcal {M}$ that are not necessarily martingale difference sequences. It is worth pointing out that the infinite sums of positive operators stated above may not always make sense as operators, but we only consider below the special cases where they do converge in the sense of the measure topology.

We will now describe various noncommutative martingale Hardy spaces associated with symmetric quasi-Banach function spaces.

Assume that E is a symmetric quasi-Banach function space. We denote by $\mathcal {F}_E$ the collection of all finite martingales in $E(\mathcal {M})$ . For $x=(x_k)_{k\geq 1} \in \mathcal {F}_E$ , we set:

$$ \begin{align*} \| x \|_{\mathcal{H}_E^c}= \| S_c(x)\|_{E(\mathcal{M})}. \end{align*} $$

Then $(\mathcal {F}_E, \|\cdot \|_{\mathcal {H}_E^c})$ is a quasi-normed space. If we denote by $(e_{i,j})_{i,j \geq 1}$ the family of unit matrices in $B(\ell _2(\mathbb {N}))$ , then the correspondence $x\mapsto \sum _{k\geq 1} dx_k \otimes e_{k,1}$ maps $\mathcal {F}_E$ isometrically into a (not necessarily closed) linear subspace of $E(\mathcal {M}\overline {\otimes } B(\ell _2(\mathbb {N})))$ . We define the column Hardy space $\mathcal {H}_E^c (\mathcal {M})$ to be the completion of $(\mathcal {F}_E, \|\cdot \|_{\mathcal {H}_E^c})$ . It then follows that $\mathcal {H}_E^c(\mathcal {M})$ embeds isometrically into a closed subspace of the quasi-Banach space $E(\mathcal {M}\overline {\otimes } B(\ell _2(\mathbb {N})))$ .

In the sequel, we will also make use of the more general space $E(\mathcal {M};\ell _2^c)$ which is defined as the set of all sequences $a=(a_k)$ in $E(\mathcal {M})$ for which $\mathcal {S}_c(a) \in E(\mathcal {M})$ . In this case, we set

$$ \begin{align*} \big\|a \big\|_{E(\mathcal{M};\ell_2^c)} = \|\mathcal{S}_c(a) \|_{E(\mathcal{M})}. \end{align*} $$

Under the above quasi-norm, one can easily see that $E(\mathcal {M};\ell _2^c)$ is a quasi-Banach space. The closed subspace of $E(\mathcal {M};\ell _2^c)$ consisting of adapted sequences will be denoted by $E^{\text {ad}}(\mathcal {M};\ell _2^c)$ . That is,

$$ \begin{align*} E^{\text{ad}}(\mathcal{M};\ell_2^c)=\big\{ (a_n)_{n\geq 1} \in E(\mathcal{M};\ell_2^c) : \forall n\geq 1, a_n \in E(\mathcal{M}_n) \big\}. \end{align*} $$

Note that for $1<p<\infty $ , it follows from the noncommutative Stein inequality that $L_p^{\text {ad}}(\mathcal {M};\ell _2^c)$ is a complemented subspace of $L_p(\mathcal {M};\ell _2^c)$ . One should not expect such complementation if one merely assume that E is a quasi-Banach symmetric function space.

Next, we will discuss conditioned versions of the spaces defined earlier. Consider the linear space $\mathcal {FS}$ consisting of all $x \in \mathcal {M}$ such that there exists a projection $e\in \mathcal {M}_1$ , $\tau (e)<\infty $ , and $x=exe$ . We should note that if $\mathcal {M}$ is finite, then $\mathcal {FS}=\mathcal {M}$ . For every $n\geq 1$ and $0<p \leq \infty $ , we define the space $L_p^c(\mathcal {M},\mathcal {E}_n)$ to be the completion of $\mathcal {FS}$ with respect to the quasi-norm:

$$ \begin{align*} \big\|x\big\|_{L_p^c(\mathcal{M},\mathcal{E}_n)} =\big\|\mathcal{E}_n(x^*x) \big\|_{p/2}^{1/2}. \end{align*} $$

We would like to emphasize here that if $x=exe \in \mathcal {FS}$ is as described above, then $\mathcal {E}_n(x^*x)=e\mathcal {E}_n(x^*x)e$ is a well-defined operator in $\mathcal {M}$ and since $\tau (e)<\infty $ , it follows that $\mathcal {E}_n(x^*x) \in L_{p/2}(\mathcal {M})$ .

According to [Reference Junge24], for every $0<p\leq \infty $ , there exists an isometric right $\mathcal {M}_n$ -module map $u_{n,p}: L_p^c(\mathcal {M}, \mathcal {E}_n) \to L_p(\mathcal {M}_n;\ell _2^c)$ such that

(2.1) $$ \begin{align} u_{n,p}(x)^* u_{n,q}(y)=\mathcal{E}_n(x^*y) \otimes e_{1,1}, \end{align} $$

whenever $x\in L_p^c(\mathcal {M};\mathcal {E}_n)$ , $y \in L_q^c(\mathcal {M};\mathcal {E}_n)$ , and $1/p +1/q \leq 1$ . An important fact about these maps is that they are independent of p as the index p in the presentation of [Reference Junge24] was only needed to accommodate the nontracial case. Below, we will simply use $u_n$ for $u_{n,p}$ .

For $0<p\leq \infty $ , the range of $u_n$ is complemented in $L_p(\mathcal {M}_n,\ell _2^c)$ . In fact, as proved in [Reference Junge24, Proposition 2.8(iii)], there exists a contractive projection $\mathcal {Q}_{n}$ from $L_p(\mathcal {M}_n; \ell _2^c)$ onto the range of $u_{n}$ such that for every $\xi \in L_p(\mathcal {M}_n;\ell _2^c)$ ,

$$ \begin{align*} \mathcal{Q}_{n}(\xi)^*\mathcal{Q}_{n}(\xi)\leq \xi^*\xi. \end{align*} $$

This fact will be used in the sequel.

Let $\mathfrak {F}$ be the collection of all finite sequences $a=(a_n)_{n\geq 1}$ in $\mathcal {FS}$ . For $0<p<\infty $ , we defined the conditioned space $L_p^{\text {cond}}(\mathcal {M}; \ell _2^c)$ to be the completion of the linear space $\mathfrak {F}$ with respect to the quasi-norm:

(2.2) $$ \begin{align} \big\| a \big\|_{L_p^{\text{cond}}(\mathcal{M}; \ell_2^c)} = \big\| \sigma_c(a)\big\|_p \end{align} $$

(here, we take $\mathcal {E}_0=\mathcal {E}_1$ ). According to [Reference Junge24], $L_p^{\text {cond}}(\mathcal {M};\ell _2^c)$ can be isometrically embedded into an $L_p$ -space associated to a semifinite von Neumann algebra by means of the following map:

$$ \begin{align*} U : L_p^{\text{cond}}(\mathcal{M}; \ell_2^c) \to L_p(\mathcal{M} \overline{\otimes} B(\ell_2(\mathbb{N}^2)))\end{align*} $$

defined by setting:

$$ \begin{align*} U((a_n)_{n\geq 1}) = \sum_{n\geq 1} u_{n-1}(a_n) \otimes e_{n,1}, \quad (a_n)_{n\geq 1} \in \mathfrak{F}. \end{align*} $$

The range of U may be viewed as a double indexed sequences $(x_{n,k})$ such that $x_{n,k} \in L_p(\mathcal {M}_n)$ for all $k\geq 1$ . As an operator affiliated with $\mathcal {M} \overline {\otimes } B(\ell _2(\mathbb {N}^2))$ , this may be expressed as $\sum _{n,k} x_{n,k} \otimes e_{k,1} \otimes e_{n,1}$ . It is immediate from (2.1) that if $(a_n)_{n\geq 1} \in \mathfrak {F}$ and $(b_n)_{n\geq 1} \in \mathfrak {F}$ , then

(2.3) $$ \begin{align} U( (a_n))^* U((b_n)) =\Big(\sum_{n\geq 1} \mathcal{E}_{n-1}(a_n^* b_n)\Big) \otimes e_{1,1} \otimes e_{1,1}. \end{align} $$

In particular, if $(a_n)_{n\geq 1} \in \mathfrak {F}$ then $\| (a_n) \|_{L_p^{\text {cond}}(\mathcal {M}; \ell _2^c)}=\|U( (a_n))\|_p$ and hence U is indeed an isometry.

Now, we generalize the notion of conditioned spaces to the setting of symmetric spaces of operators. This is done in steps.

$\bullet $ Assume first that E is a symmetric quasi-Banach function space satisfying $L_p \cap L_\infty \subseteq E\subseteq L_p +L_\infty $ for some $0<p<\infty $ and $L_p \cap L_\infty $ is dense in E. This is the case for instance when E is a separable fully symmetric quasi-Banach function space. For a given sequence $a=(a_n)_{n\geq 1} \in \mathfrak {F}$ , we set:

$$ \begin{align*} \big\| (a_n) \big\|_{E^{\text{cond}}(\mathcal{M}; \ell_2^c)} = \big\| \sigma_c(a)\big\|_{E(\mathcal{M})}=\big\| U( (a_n))\big\|_{E(\mathcal{M} \overline{\otimes} B(\ell_2(\mathbb{N}^2)))}. \end{align*} $$

This is well-defined and induces a quasi-norm on the linear space $\mathfrak {F}$ . We define the quasi-Banach space $E^{\text {cond}}(\mathcal {M}; \ell _2^c)$ to be the completion of the quasi normed space $(\mathfrak {F}, \|\cdot \|_{E^{\text {cond}}(\mathcal {M}; \ell _2^c)} )$ . The space $E^{\text {cond}}(\mathcal {M};\ell _2^c)$ will be called the column conditioned space associated with E. It is clear that U extends to an isometry from $E^{\text {cond}}(\mathcal {M};\ell _2^c)$ into $E(\mathcal {M} \overline {\otimes } B(\ell _2(\mathbb {N}^2)))$ which we will still denote by U.

Below, we use the notation $E^{0, \textrm {cond}}(\mathcal {M}; \ell _2^c)$ for the closure of the linear space $\mathfrak {F}^0=\{a=(a_n) \in \mathfrak {F}: a_1=0\}$ in $E^{\text {cond}}(\mathcal {M}; \ell _2^c)$ . One can easily see that for $a=(a_n)_{n\geq 1} \in \mathfrak {F}$ , we have

$$ \begin{align*} \max\Big\{ \big\| \mathcal{E}_1(|a_1|^2)^{1/2} \big\|_{E(\mathcal{M}_1)}, \big\| (a_n)_{n\geq 2}\big\|_{E^{\text{cond}}(\mathcal{M};\ell_2^c)} \Big\} \leq \big\| (a_n)_{n\geq 1}\big\|_{E^{\text{cond}}(\mathcal{M};\ell_2^c)}. \end{align*} $$

This shows that we have the direct sum

(2.4) $$ \begin{align} E^{\text{cond}}(\mathcal{M};\ell_2^c)= E^c(\mathcal{M};\mathcal{E}_1) \oplus E^{0, \textrm{cond}}(\mathcal{M};\ell_2^c). \end{align} $$

$\bullet $ Assume now that $E\subseteq L_p +L_q$ for some $0<p,q<\infty $ that is not necessarily separable. We set

$$ \begin{align*} E^{\text{cond}}(\mathcal{M}; \ell_2^c)=\big\{ x \in (L_p +L_q)^{\text{cond}}(\mathcal{M};\ell_2^c) : U(x) \in E(\mathcal{M} \overline{\otimes} B(\ell_2(\mathbb N^2)) )\big\} \end{align*} $$

equipped with the quasi-norm:

$$ \begin{align*} \big\|x \big\|_{E^{\text{cond}}(\mathcal{M}; \ell_2^c)} =\big\| U(x) \big\|_{E(\mathcal{M} \overline{\otimes} B(\ell_2(\mathbb{N}^2)))}. \end{align*} $$

It is clear that $E^{\text {cond}}(\mathcal {M}; \ell _2^c)$ as defined is a linear quasi-normed space. We claim that it is complete. Indeed, if $(x_\nu )_{\nu \geq 1}$ is a Cauchy sequence in $E^{\text {cond}}(\mathcal {M}; \ell _2^c)$ , then it converges to some $x \in (L_p +L_q)^{\text {cond}}(\mathcal {M};\ell _2^c)$ . Since $( U(x_\nu ))_{\nu \geq 1}$ is also a Cauchy sequence in the quasi-Banach space $E(\mathcal {M} \overline {\otimes } B(\ell _2(\mathbb {N}^2)))$ and $E(\mathcal {M} \overline {\otimes } B(\ell _2(\mathbb {N}^2))) \subseteq (L_p +L_q)(\mathcal {M} \overline {\otimes } B(\ell _2(\mathbb {N}^2)))$ , it follows that $x \in E^{\text { cond}}(\mathcal {M}; \ell _2^c)$ and $(x_\nu )_{\nu \geq 1}$ converges to x in $E^{\text {cond}}(\mathcal {M}; \ell _2^c)$ . We should note here that if $L_p \cap L_\infty $ is dense in E then the above definition coincides with the one described in the previous bullet. We also define

$$ \begin{align*} E^{0, \textrm{cond}}(\mathcal{M}; \ell_2^c)=(L_p +L_q)^{0, \textrm{cond}}(\mathcal{M}; \ell_2^c) \cap E^{\text{cond}}(\mathcal{M}; \ell_2^c). \end{align*} $$

The direct sum stated in (2.4) still applies.

We now recall the construction of column conditioned martingale Hardy spaces. As in the conditioned spaces, we describe the noncommutative conditioned Hardy spaces in steps. Let $\mathfrak {F}(M)$ be the collection of all finite martingale $(x_n)_{1\leq n \leq N}$ for which $ x_N \in \mathcal {FS}$ . We can easily see that for every $0<p\leq \infty $ , $\mathfrak {F}(M) \subseteq \mathsf {h}_p^c(\mathcal {M})$ . As in the case of conditioned spaces, $\mathfrak {F}(M)$ is dense in $\mathsf {h}_p^c(\mathcal {M})$ when $0<p<\infty $ .

$\bullet $ First, assume that $E\subseteq L_2 + L_\infty $ . In this case, column conditioned square functions are well-defined for bounded martingales in $E(\mathcal {M})$ . We define $\mathsf {h}_E^c(\mathcal {M})$ to be the collection of all bounded martingale x in $E(\mathcal {M})$ for which $s_c(x) \in E(\mathcal {M})$ . We equip $\mathsf {h}_E^c(\mathcal {M})$ with the norm:

$$ \begin{align*} \big\|x \big\|_{\mathsf{h}_E^c} = \big\|s_c(x)\big\|_{E(\mathcal{M})}. \end{align*} $$

One can easily verify that $(\mathsf {h}_E^c(\mathcal {M}), \|\cdot \|_{\mathsf {h}_E^c})$ is complete.

$\bullet $ Next, we consider quasi-Banach space E such that $L_p\cap L_\infty $ is dense in E for some $0<p<\infty $ . This is the case if E is separable. Let $x \in \mathfrak {F}(M)$ . As noted above, $s_c(x) \in L_p(\mathcal {M}) \cap \mathcal {M}$ . In particular, $s_c(x) \in E(\mathcal {M})$ . We equip $\mathfrak {F}(M)$ with the quasi-norm

$$ \begin{align*} \big\|x \big\|_{\mathsf{h}_E^c} = \big\|s_c(x)\big\|_{E(\mathcal{M})}=\big\|(dx_n)\big\|_{E^{\mathrm{cond}}(\mathcal{M};\ell_2^c)}. \end{align*} $$

The column conditioned Hardy space $\mathsf {h}_E^c(\mathcal {M})$ is the completion of $(\mathfrak {F}(M), \|\cdot \|_{\mathsf {h}_E^c})$ . Clearly, the map $x\mapsto (dx_n)$ (from $\mathfrak {F}(M)$ into $\mathfrak {F}$ ) extends to an isometry from $\mathsf {h}_E^c(\mathcal {M})$ into $E^{\mathrm {cond}}(\mathcal {M};\ell _2^c)$ which we denote by $\mathcal {D}_c$ . In particular, $\mathsf {h}_E^c(\mathcal {M})$ is isometrically isomorphic to a subspace of $E(\mathcal {M}\overline {\otimes }B(\ell _2(\mathbb {N}^2)))$ via the isometry $U\mathcal {D}_c$ . We should note here that if $L_2 \cap L_\infty $ is dense in E and $E\subseteq L_2 +L_\infty $ , then the two definitions provide the same space.

$\bullet $ Assume now that $E \subset L_q + L_q$ for $0<p,q<\infty $ . As in the case of conditioned spaces, we set

$$ \begin{align*} \mathsf{h}_E^c(\mathcal{M})=\big\{ x \in \mathsf{h}_{L_p +L_q}^c(\mathcal{M}) : U\mathcal{D}_c(x) \in E(\mathcal{M} \overline{\otimes} B(\ell_2(\mathbb N^2)) )\big\} \end{align*} $$

equipped with the quasi-norm:

$$ \begin{align*} \big\|x \big\|_{\mathsf{h}_E^c} =\big\| U\mathcal{D}_c(x) \big\|_{E(\mathcal{M} \overline{\otimes} B(\ell_2(\mathbb{N}^2)))}. \end{align*} $$

Since the operator U is independent of the index, one can easily see that the space $\mathsf {h}_E^c(\mathcal {M})$ is independent of p and q. Moreover, one can verify as in the case of conditioned spaces that the quasi-normed space $(\mathsf {h}_E^c(\mathcal {M}), \|\cdot \|_{\mathsf {h}_E^c})$ is complete. Furthermore, if E is such that $L_2 \cap L_\infty $ is dense in E then $\mathsf {h}_E^c(\mathcal {M})$ coincides with the one defined through completion considered in the second bullet.

The following fact will be used in the sequel.

Proof Let $n\geq 1$ and $b \in \mathcal {FS}$ . Using the Kadison–Schwarz inequality for conditional expectations, we have

(2.5) $$ \begin{align} \mathcal{E}_{n-1}(|\mathcal{E}_n(b)-\mathcal{E}_{n-1}(b)|^2) \leq \mathcal{E}_{n-1}(|b|^2). \end{align} $$

We define $\Pi : \mathfrak {F} \to \mathsf {h}_E^c(\mathcal {M})$ by setting:

$$ \begin{align*} a=(a_n)_{n\geq 1} \mapsto \sum_{n\geq 1} \mathcal{E}_n(a_n)- \mathcal{E}_{n-1}(a_n). \end{align*} $$

It follows from (2.5) that $s_c(\Pi (a)) \leq \sigma _c(a)$ for every $a\in \mathfrak {F}$ . If $\mathfrak {F}$ is dense in $E^{\mathrm {cond}}(\mathcal {M};\ell _2^c)$ , then $\Pi $ extends to a bounded linear map from $E^{\mathrm {cond}}(\mathcal {M};\ell _2^c)$ onto $\mathsf {h}_E^c(\mathcal {M})$ with $\|\Pi : E^{\mathrm {cond}}(\mathcal {M};\ell _2^c) \to \mathsf {h}_E^c(\mathcal {M})\| \leq 1$ . Clearly, if $x \in \mathsf {h}_E^c(\mathcal {M})$ , we have $\Pi {\mathcal D}_c (x)=x$ . This verifies the lemma for the case where E is separable.

For the general case, we observe that from the fact that $\mathfrak {F}$ is dense in $(L_p +L_q)^{\mathrm {cond}}(\mathcal {M};\ell _2^c)$ , the inequality on conditioned square functions above can be restated as:

(2.6) $$ \begin{align} \big| U{\mathcal D}_c(\Pi( y)) \big| \leq \big| U( y)\big|, \quad y\in (L_p + L_q)^{\mathrm{cond}}(\mathcal{M};\ell_2^c). \end{align} $$

If $y \in E^{\mathrm {cond}}(\mathcal {M};\ell _2^c)$ , then by definition $y \in (L_p +L_q)^{\mathrm {cond}}(\mathcal {M};\ell _2^c)$ and $U(y) \in E(\mathcal {M} \overline {\otimes } B(\ell _2(\mathbb {N}^2)))$ . By the separable case, $\Pi (y) \in \mathsf {h}_{L_p +L_q}^c(\mathcal {M})$ . Moreover, it follows from (2.6) that $U{\mathcal D}(\Pi (y)) \in E(\mathcal {M} \overline {\otimes } B(\ell _2(\mathbb {N}^2)))$ . This means, $\Pi (y)\in \mathsf {h}_E^c(\mathcal {M})$ with $\|\Pi (y)\|_{\mathsf {h}_E^c}\leq \| y\|_{E^{\mathrm {cond}}(\mathcal {M};\ell _2^c)}$ .  ▪

We refer to [Reference Cadilhac and Ricard6, Reference Dirksen11, Reference Jiao19, Reference Jiao, Sukochev, Zanin and Zhou22, Reference Junge and Xu26, Reference Randrianantoanina and Wu42, Reference Randrianantoanina, Wu and Xu43] for more information on noncommutative Hardy spaces associated with symmetric spaces of measurable operators. In the sequel, noncommutative column Hardy spaces associated with Orlicz space $L_\Phi $ will be denoted by $\mathcal {H}_\Phi ^c(\mathcal {M})$ and $\mathsf {h}_\Phi ^c(\mathcal {M})$ while those associated with the Lorentz space $L_{p,q}$ will be denoted by $\mathcal {H}_{p,q}^c(\mathcal {M})$ and $\mathsf {h}_{p,q}^c(\mathcal {M})$ .

We conclude this subsection with a description of the dual space of the Hardy space $\mathsf {h}_1^c(\mathcal {M})$ . A martingale x belongs to the column little bmo space denoted by $\mathsf {bmo}^c(\mathcal {M})$ if

$$ \begin{align*} \big\| x \big\|_{\mathsf{bmo}^c}= \max\Big\{ \big\|x_1\big\|_\infty; \mathop{\mathrm{sup}}_{m} \mathop{\mathrm{sup}}_{1\leq n\leq m }\big\| \mathcal{E}_n(|x_m-x_n|^2) \big\|_\infty^{1/2} \Big\} <\infty. \end{align*} $$

The Banach space $(\mathsf {bmo}^c(\mathcal {M}), \|\cdot \|_{\mathsf {bmo}^c})$ was introduced in [Reference Perrin33] for finite case where it was shown that it coincides with the dual of the Hardy space $\mathsf {h}_1^c(\mathcal {M})$ . The proof given there can be easily generalized to the semifinite case. We record this for further use:

(2.7) $$ \begin{align} (\mathsf{h}_1^c(\mathcal{M}))^* =\mathsf{bmo}^c(\mathcal{M}) \end{align} $$

with equivalent norms.

2.3 Atomic decompositions for martingale Orlicz–Hardy spaces

In this subsection, we will analyze conditioned spaces and conditioned Hardy spaces associated with Orlicz spaces. More specifically, we will describe a type of atomic decomposition in the context of Orlicz spaces. Results from this subsection play key role in the next section. Toward this end, we will start from setting up some notations.

Throughout this subsection, we always assume that $0<p \leq q <2$ and $\Phi $ is an Orlicz function that is p-convex and q-concave. First, we fix a positive Borel measure $\mu $ on the interval $[0,\infty )$ so that:

(2.8) $$ \begin{align} \Phi(t)\approx_{p,q}\int_0^\infty \min\{(ts)^p, (ts)^q\} \ d\mu(s). \end{align} $$

The existence of such integral representation was proved for convex functions (see the proof of [Reference Jiao, Sukochev and Zanin21, Lemma 6.2]). The argument given in [Reference Jiao, Sukochev and Zanin21] can be readily adjusted to include the more general case of p-convex functions when $0<p<1$ .

Next, we fix an Orlicz function $\Theta $ so that $L_\Phi $ admits the factorization:

(2.9) $$ \begin{align} L_\Phi=L_2 \odot L_\Theta, \end{align} $$

where the product $L_2 \odot L_\Theta $ is the collection of all function f that admit factorization $f=gh$ with $g\in L_2$ and $h\in L_{\Theta }$ . The quasi-norm on $L_2\odot L_\Theta $ is given by:

$$ \begin{align*} \big\|f\big\|_{L_2 \odot L_\Theta} :=\inf\left\{\big\|g\big\|_{L_2}\big\|h \big\|_{L_\Theta}: g\in L_2, h\in L_\Theta, f=gh\right\}. \end{align*} $$

We note that the Orlicz function $\Theta $ can be taken to be the inverse of the function $t\mapsto t^{-1/2}\Phi ^{-1}(t)$ for $t>0$ . We refer to [Reference Maligranda31] for more details.

We will also make use of the following function:

(2.10) $$ \begin{align} \Psi(t)= \int_0^\infty (t^{2-p}s^{-p} + t^{2-q}s^{-q})^{-1}\ d\mu(s), \end{align} $$

where $\mu $ is the positive Borel measure from the representation of $\Phi $ in (2.8). The reason for the consideration of $\Psi $ is summarized in the next lemma. The first three items are straightforward generalizations of [Reference Randrianantoanina, Wu and Zhou44, Proposition 3.3], whereas the last item can be deduced as in [Reference Randrianantoanina, Wu and Zhou44, Lemma 3.4].

We now introduce a concept of atoms for conditioned space constructed from the Orlicz function space $L_\Phi $ .

The above definition was motivated by the case of $L_r^{\mathrm {cond}}(\mathcal {M};\ell _2^c)$ for $1\leq r<2$ introduced for the first time in [Reference Perrin34] and explored further in [Reference Hong, Junge and Parcet17]. We also refer to [Reference Chen, Randrianantoanina and Xu7] where the same concept was considered for the case of conditioned Hardy space $\mathsf {h}_r^c(\mathcal {M})$ for the range $0<r\leq 1$ . More recently, the case of Orlicz-conditioned Hardy space $\mathsf {h}_\varphi ^c(\mathcal {M})$ associated with convex Orlicz function $\varphi $ was formulated in [Reference Randrianantoanina, Wu and Zhou44] in the context of $\varphi $ -moments.

We should note that since strictly lower triangular matrices were used in the definition of algebraic atoms, it follows that if $x=(x_n)_{n\geq 1}$ is an algebraic $L_\Phi ^{c, \textrm {cond}}$ -atom then $x_1=0$ .

The next lemma can be deduced as in the first part of [Reference Perrin34, Theorem 3.6.10] using the factorization $L_\Phi =L_2 \odot L_\Theta $ . We include the argument for the sake of completeness.

Proof Let $x=\alpha \ .\ \beta $ be an algebraic $L_\Phi ^{c, \textrm {cond}}$ -atom. As observed earlier, $x_1=0$ and for $n\geq 2$ , $x_n= \sum _{j<n} \alpha _{n,j} \beta _j$ . Since $\beta $ is adapted, it follows that

$$ \begin{align*} \mathcal{E}_{n-1}|x_n|^2 =\sum_{m,j<n} \beta_m^* \mathcal{E}_{n-1}(\alpha_{n,m}^* \alpha_{n,j}) \beta_j. \end{align*} $$

From the property of the module map $u_{n-1}$ , we have

$$ \begin{align*} \mathcal{E}_{n-1}|x_n|^2 \otimes e_{1,1} &= \sum_{m,j <n} (\beta_m^* \otimes e_{1,1}). u_{n-1}(\alpha_{n,m})^* u_{n-1}(\alpha_{n,j}). (\beta_j \otimes e_{1,1}) \\ &=| \sum_{j<n} u_{n-1}(\alpha_{n,j}).(\beta_j \otimes e_{1,1})|^2. \end{align*} $$

This implies that

$$ \begin{align*} \sigma_c^2(x) \otimes e_{1,1} =\sum_{n\geq 2} | \sum_{j<n} u_{n-1}(\alpha_{n,j}).(\beta_j \otimes e_{1,1})|^2. \end{align*} $$

We write further that

$$ \begin{align*} \sigma_c^2(x) \otimes e_{1,1} \otimes e_{1,1} =| \sum_{n\geq 2} \sum_{j<n} [u_{n-1}(\alpha_{n,j}).(\beta_j \otimes e_{1,1})] \otimes e_{n,1} |^2. \end{align*} $$

This allows us to deduce that

$$ \begin{align*} \sigma_c(x) \otimes e_{1,1} \otimes e_{1,1} = \big|\big(u_{n-1}(\alpha_{n,j}) \big)_{j<n}. \ \sum_{k\geq 1} \beta_k \otimes e_{1,1} \otimes e_{k,1} \big|, \end{align*} $$

where the strictly lower triangular matrix $\widehat {\alpha }=(\big (u_{n-1}(\alpha _{n,j}) )_{j<n}$ takes its values in $L_2(\mathcal {M} \overline {\otimes } B(\ell _2(\mathbb {N})))$ . We may view $\widehat {\alpha }$ as an operator affiliated with $\mathfrak {M}=\mathcal {M} \overline {\otimes } B(\ell _2(\mathbb {N}))) \overline {\otimes } B(\ell _2(\mathbb {N})))$ . If $\mu (\cdot )$ denote the generalized singular number relative to $\mathfrak {M}$ equipped with its natural trace, then

$$ \begin{align*} \big\| \sigma_c(x)\big\|_{L_\Phi(\mathcal{M})} = \|\mu(\sigma_c(x) \otimes e_{1,1} \otimes e_{1,1}) \big\|_{L_\Phi}. \end{align*} $$

It follows from [Reference Fack and Kosaki12, Theorem 4.2] that

$$ \begin{align*} \big\| \sigma_c(x)\big\|_{L_\Phi(\mathcal{M})} &\leq \|\mu(\widehat{\alpha}) \big\|_{2} .\big\| \mu\big( \sum_{k\geq 1} \beta_k \otimes e_{1,1} \otimes e_{k,1}\big) \big\|_{L_\Theta}\\ &= \big\| \widehat{\alpha}\big\|_{2}. \big\| \big(\sum_{k\geq 1} |\beta_k|^2\big)^{1/2} \big\|_{L_\Theta}\\ &\leq \big(\sum_{j<n} \| \alpha_{n,j}\|_2^2 \big)^{1/2}. \big\| \big(\sum_{k\geq 1} |\beta_k|^2\big)^{1/2} \big\|_{L_\Theta}. \end{align*} $$

This proves that $\| x \|_{L_\Phi ^{\mathrm {cond}}(\mathcal {M};\ell _2^c)}\leq 1$ .  ▪

Using the above notion of atoms, we may naturally consider the following concept of atomic decompositions:

Definition 2.7 A sequence $x \in L_\Phi (\mathcal {M};\ell _2^c)$ is said to admit an algebraic $L_\Phi ^{c, \textrm {cond}}$ -atomic decomposition if

$$ \begin{align*} x =\sum_{k} \lambda_k a^{(k)}, \end{align*} $$

where for each k, $a^{(k)}$ is either an algebraic $L_\Phi ^{c, \textrm {cond}}$ -atom or $a^{(k)}$ belongs to the unit ball of the conditioned space $L_\Phi ^c(\mathcal {M};\mathcal {E}_1)$ and $\lambda _k \in \mathbb {C}$ satisfying $\sum _k|\lambda _k|^p <\infty $ for $0<p<1$ and $\sum _k|\lambda _k| <\infty $ for $1\leq p<2$ . Since $\Phi $ is p-convex and algebraic $L_\Phi ^{c, \textrm {cond}}$ -atoms belong to the unit ball of $L_\Phi ^{\mathrm {cond}}(\mathcal {M};\ell _2^c)$ , it follows that if x admits an algebraic $L_\Phi ^{c, \textrm {cond}}$ -atomic decomposition then it belongs to $L_\Phi ^{\mathrm {cond}}(\mathcal {M};\ell _2^c)$ .

Following [Reference Perrin34], the corresponding algebraic atomic column conditioned space $L_{\Phi ,\mathrm {aa}}^{ \textrm {cond}}(\mathcal {M};\ell _2^c)$ is defined to be the completion of the space of all x that admit algebraic $L_\Phi ^{c, \textrm {cond}}$ -atomic decompositions in the space $L_\Phi ^{\mathrm {cond}}(\mathcal {M};\ell _2^c)$ . If x admits an algebraic $L_\Phi ^{c, \textrm {cond}}$ -atomic decomposition, we set:

$$ \begin{align*} \|x\|_{L_{\Phi,\mathrm{aa}}^{\mathrm{cond}}(\mathcal{M};\ell_2^c)}=\inf\Big(\sum_k |\lambda_k|^p\Big)^{1/p} \quad \text{for} \ 0<p\leq 1\end{align*} $$

and

$$ \begin{align*} \|x\|_{L_{\Phi,\mathrm{aa}}^{\mathrm{cond}}(\mathcal{M};\ell_2^c)}=\inf\sum_k |\lambda_k| \quad \text{for} \ 1<p<2,\end{align*} $$

where the infimum is taken over all decompositions of x as described above. We refer the reader to [Reference Chen, Randrianantoanina and Xu7] for a more in-depth discussion on the need to separate the two cases.

The following result generalizes the atomic decomposition of conditioned $L_p$ -spaces from [Reference Perrin34] in two directions: it is valid for Orlicz spaces and also cover the quasi-Banach space range. This will play a crucial role in the next section. The approach of [Reference Perrin34] was by duality which is not applicable to the present situation since we are dealing with not necessarily convex functions. Our constructive proof given below combined ideas from [Reference Chen, Randrianantoanina and Xu7] and the case of Hardy spaces associated with convex Orlicz functions considered in [Reference Randrianantoanina, Wu and Zhou44]. This may be of independent interest.

Theorem 2.8 Let $\Phi $ be p-convex and q-concave for $0<p\leq q<2$ . Then the two spaces $ L_{\Phi ,\mathrm {aa}}^{ \textrm {cond}}(\mathcal {M};\ell _2^c)$ and $ L_\Phi ^{\mathrm {cond}}(\mathcal {M};\ell _2^c)$ coincide (with constant of isomorphism depending only on p and q).

More precisely, every $ x \in \mathfrak {F}^0$ admits a factorization $x= \alpha \, .\, \beta $ where $\alpha $ is a strictly lower triangular matrix in $L_2(\mathcal {M} \overline {\otimes } B(\ell _2(\mathbb {N})))$ and $\beta \in L_\Theta ^{\mathrm {ad}}(\mathcal {M},\ell _2^c)$ satisfying:

$$ \begin{align*} \big\| \alpha\big\|_2 \, .\, \big\| \beta\big\|_{L_\Theta(\mathcal{M};\ell_2^c)} \lesssim_{p,q} \big\| x \big\|_{L_\Phi^{\mathrm{cond}}(\mathcal{M};\ell_2^c)}. \end{align*} $$

Proof First, we recall from earlier discussion that $L_{\Phi ,\mathrm {aa}}^{ \textrm {cond}}(\mathcal {M};\ell _2^c) \subseteq L_\Phi ^{\mathrm {cond}}(\mathcal {M};\ell _2^c)$ . Thus, we only need to verify one inclusion. This will be deduced from the second part of the theorem.

Let $ x=(x_n)_{n=1}^N \in \mathfrak {F}^0$ . The construction below is an adaptation of the argument used in [Reference Randrianantoanina, Wu and Zhou44]. By definition, $x_1=0$ and there exists a projection $e\in \mathcal {M}_1$ with $\tau (e)<\infty $ such that for $2\leq n\leq N$ , $x_n=ex_n e$ .

First, we note that for $j\geq 2$ , we have $\sigma _{c,j}(x) \in e\mathcal {M}_{j-1}e$ . By approximation, we may assume that each of the $\sigma _{c,j}(x)$ s is invertible with bounded inverse in $e\mathcal {M} e$ . Below, we simply write $\sigma _j$ for $\sigma _{c,j}(x)$ and the function $\Psi $ is as defined in (2.10). Let $\lambda>0$ to be determined later. For $n\geq 2$ , we write

$$ \begin{align*} x_n &=x_n \Psi(\lambda\sigma_n)\Psi(\lambda\sigma_n)^{-1}\\ &=x_n\Psi(\lambda\sigma_n)\big[ \Psi(\lambda\sigma_2)^{-1} + \sum_{3\leq m\leq n} \Psi(\lambda\sigma_m)^{-1}-\Psi(\lambda\sigma_{m-1})^{-1}\big]\\ &=x_n \Psi(\lambda\sigma_n)\Psi(\lambda\sigma_2)^{-1} + \sum_{2\leq j<n} x_n \Psi(\lambda\sigma_n)\big(\Psi(\lambda\sigma_{j+1})^{-1} -\Psi(\lambda\sigma_j)^{-1}\big). \end{align*} $$

We define the strictly lower triangular matrix $\alpha $ by setting:

(2.11) $$ \begin{align} \begin{cases} \alpha_{n,1} &:=x_n\Psi(\lambda\sigma_n)\Psi(\lambda\sigma_2)^{-1/2};\\ \alpha_{n,j}&:=\displaystyle{x_n\Psi(\lambda\sigma_n)\left(\Psi(\lambda\sigma_{j+1})^{-1}-\Psi(\lambda\sigma_{j})^{-1}\right)^{1/2},\quad\text{for }2 \leq j<n.} \end{cases} \end{align} $$

We should point out here that since the function $t\mapsto \Psi (t^{1/2})$ is operator monotone decreasing and $\lambda ^2\sigma _{j}^2 \leq \lambda ^2 \sigma _{j+1}^2$ , we have $\Psi (\lambda \sigma _{j+1}) \leq \Psi (\lambda \sigma _{j})$ . Taking inverses, $\Psi (\lambda \sigma _{j+1})^{-1}-\Psi (\lambda \sigma _{j})^{-1}\geq 0$ . Thus, taking $1/2$ -power in the expression above is justified.

The column sequence is defined by setting:

(2.12) $$ \begin{align} \begin{cases} \beta_{1,1} &:=\displaystyle{\Psi(\lambda\sigma_2)^{-1/2};}\\ \beta_{m,1}&:=\displaystyle{\left(\Psi(\lambda\sigma_{m+1})^{-1}-\Psi(\lambda\sigma_{m})^{-1}\right)^{1/2}, \quad \text{for }m\geq 2}. \end{cases} \end{align} $$

Then $\alpha =(\alpha _{n,j})_{j<n}$ is a strictly lower triangular matrix and $\beta =(\beta _{m,1})_{m\geq 1}$ is an adapted sequence. Moreover, for every $n\geq 2$ , it clearly follows from the definition that

$$ \begin{align*} (\alpha.\beta)_{n,1}= \sum_{j\geq 1} \alpha_{n,j}. \beta_{j,1}=x_n. \end{align*} $$

That is, we have the factorization $x= \alpha \, .\, \beta $ . We claim that the product $\alpha \,. \, \beta $ satisfies the desired norm estimate. We begin with the $L_2$ -norm of $\alpha $ .

$$ \begin{align*} \big\|\alpha\big\|_2^2 &=\sum_{n\geq 2} \sum_{1\leq j<n} \big\|\alpha_{n,j}\big\|_2^2\\ &=\sum_{n\geq 2} \tau\big(\Psi(\lambda\sigma_n) |x_n|^2 \Psi(\lambda\sigma_n) \big[ \Psi(\lambda\sigma_2)^{-1} +\sum_{2\leq j<n} \Psi(\lambda\sigma_{j+1})^{-1}-\Psi(\lambda\sigma_j)^{-1}\big]\big)\\ &=\sum_{n\geq 2}\tau\big(|x_n|^2\Psi(\lambda\sigma_n)\big). \end{align*} $$

Since $(\Psi (\lambda \sigma _n))$ is a predictable sequence, we have

$$ \begin{align*} \big\|\alpha\big\|_2^2&=\sum_{n\geq 2} \tau\big(\mathcal{E}_{n-1}(|x_n|^2) \Psi(\lambda\sigma_n) \big)\\ &=\sum_{n\geq 2} \tau\big( (\sigma_n^2- \sigma_{n-1}^2)\Psi(\lambda\sigma_n)\big)\\ &=\lambda^{-2}\sum_{n\geq 2} \tau\big( ((\lambda\sigma_n)^2-(\lambda \sigma_{n-1})^2)\Psi(\lambda\sigma_n)\big). \end{align*} $$

We may deduce from Lemma 2.4(iv) that there is a constant $C_{p,q}$ so that

(2.13) $$ \begin{align} \big\|\alpha\big\|_2^2 \leq C_{p,q} \lambda^{-2} \tau\big(\Phi(\lambda\sigma_c(x))\big). \end{align} $$

On the other hand, Lemma 2.4(ii) implies that there exists a constant $C_{p,q}'$ so that

(2.14) $$ \begin{align} \tau\Big[ \Theta\Big( \big(\sum_{m\geq 1} |\beta_{m,1}|^2\big)^{1/2} \Big)\Big] =\tau \Big[ \Theta\Big( \Psi (\lambda\sigma_c(x))^{-1/2} \Big)\Big] \leq C_{p,q}'\tau\big(\Phi(\lambda\sigma_c(x))\big). \end{align} $$

Let $K_{p,q}=\max \{C_{p,q}, C_{p,q}'\} +1$ . Since $t\mapsto t^{-p}\Phi (t)$ is nondecreasing, one can easily verify that $\Phi (t)\leq K_{p,q}^{-1} \Phi (K_{p,q}^{1/p}t)$ . Using this fact, we get from (2.13) and (2.14) that

$$ \begin{align*} \big\|\alpha\big\|_2^2 \leq \lambda^{-2} \tau\big(\Phi( K_{p,q}^{1/p}\lambda\sigma_c(x))\big) \end{align*} $$

and

$$ \begin{align*} \tau\left[ \Theta\left( \left(\sum_{m\geq 1} |\beta_{m,1}|^2\right)^{1/2} \right)\right] \leq \tau\big(\Phi( K_{p,q}^{1/p}\lambda\sigma_c(x))\big). \end{align*} $$

Choose $\lambda $ so that $\tau \big (\Phi ( K_{p,q}^{1/p}\lambda \sigma _c(x))\big )\leq 1$ . This can be achieved with $\lambda ^{-1}= K_{p,q}^{1/p} \big \|\sigma _c(x) \big \|_{L_\Phi (\mathcal {M})}$ . With the above choice of $\lambda $ , we clearly have $ \big \| \alpha \big \|_2 \leq K_{p,q}^{1/p} \big \| x \big \|_{L_\Phi ^{\mathrm {cond}}(\mathcal {M};\ell _2^c)}$ and $\big \|\beta \big \|_{L_\Theta (\mathcal {M};\ell _2^c)} \leq 1$ . This proves the desired estimate and therefore the second part of the theorem.

To conclude the proof, we apply the case of $\mathfrak {F}^0$ with the direct sum (2.4). We see that every $y \in \mathfrak {F}$ admits an algebraic $L_\Phi ^{c, \textrm {cond}}$ -atomic decomposition with $\|y\|_{L_{\Phi ,\mathrm {aa}}^{\mathrm {cond}}(\mathcal {M};\ell _2^c)} \lesssim _{p,q} \|y\|_{L_\Phi ^{\mathrm {cond}}(\mathcal {M};\ell _2^c)}$ . Since $\mathfrak {F}$ is dense in $L_\Phi ^{\mathrm {cond}}(\mathcal {M};\ell _2^c)$ , it follows that $L_{\Phi ,\mathrm {aa}}^{\mathrm {cond}}(\mathcal {M};\ell _2^c)=L_\Phi ^{\mathrm {cond}}(\mathcal {M};\ell _2^c)$ .  ▪

Remark 2.9 Using $\lambda =1$ in the proof above, we also obtain a moment version of the preceding theorem: given a sequence $x=(x_n)_{n\geq 1} \in \mathfrak {F}^0$ , the column matrix $\overline {x}=\sum _{n\geq 1} x_n \otimes e_{n,1}$ admits a factorization $\overline {x}=\alpha \ .\ \beta $ with $\alpha $ is a strictly lower triangular in $L_2(\mathcal {M} \overline {\otimes } B(\ell _2(\mathbb N)))$ and $\beta $ is an adapted column matrix satisfying

$$ \begin{align*} \big\|\alpha\big\|_2 + \tau\left[\Theta\left( \left(\sum_{n\geq 1} |\beta_{n,1}|^2 \right)^{1/2}\right)\right] \lesssim_{p,q} \tau\big[\Phi\big( \sigma_c(x)\big)\big]. \end{align*} $$

We consider now the version of algebraic atomic decomposition for martingale Hardy spaces. The next consideration generalizes notions from [Reference Chen, Randrianantoanina and Xu7, Reference Randrianantoanina, Wu and Zhou44] for non-necessarily convex Orlicz functions.

Following [Reference Chen, Randrianantoanina and Xu7, Reference Randrianantoanina, Wu and Zhou44], this concept of atoms naturally leads to the consideration of the corresponding atomic decomposition for conditioned Orlicz–Hardy spaces: we say that an operator $x \in L_\Phi (\mathcal {M})$ admits an algebraic $\mathsf {h}_\Phi ^c$ -atomic decomposition if

$$ \begin{align*} x =\sum_{k} \lambda_k a_k, \end{align*} $$

where for each k, $a_k$ is an algebraic $\mathsf {h}_\Phi ^c$ -atom or an element of the unit ball of $L_\Phi (\mathcal {M}_1)$ , and $\lambda _k \in \mathbb {C}$ satisfying $\sum _k|\lambda _k|^p <\infty $ for $0<p \leq 1$ and $\sum _k|\lambda _k|<\infty $ for $1<p<2$ . The corresponding algebraic atomic column martingale Hardy space $\mathsf {h}_{\Phi ,\mathrm {aa}}^c(\mathcal {M})$ is defined to be the space of all x which admit a algebraic $\mathsf {h}_\Phi ^c$ -atomic decomposition and is equipped with

$$ \begin{align*} \|x\|_{\mathsf{h}_{\Phi,\mathrm{aa}}^c}=\inf\Big(\sum_k |\lambda_k|^p\Big)^{1/p} \quad \text{for }0<p\le1 \end{align*} $$

and

$$ \begin{align*} \|x\|_{\mathsf{h}_{\Phi,\mathrm{aa}}^c}=\inf\,\sum_k |\lambda_k| \quad \text{for }1< p<2, \end{align*} $$

where the infimum are taken over all decompositions of x as described above.

We note that the above concepts were introduced in [Reference Randrianantoanina, Wu and Zhou44] for the case where $\Phi $ is a convex function. Our main focus here is the case where $\Phi $ is p-convex with $0<p<1$ .

The next result is an extension of [Reference Chen, Randrianantoanina and Xu7, Theorem 3.10] to the case of Orlicz function spaces. It follows immediately from Theorem 2.8 and the complementation result stated in Lemma 2.3.

Corollary 2.11 Let $0<p<q<2$ and $\Phi $ is an Orlicz function that is p-convex and q-concave. Then

$$ \begin{align*} \mathsf{h}_\Phi^{c}(\mathcal{M})= \mathsf{h}_{\Phi,\mathrm{aa}}^c(\mathcal{M}). \end{align*} $$

More precisely, if $ x \in \mathsf {h}_\Phi ^c(\mathcal {M})$ , then x admits a unique decomposition $x=x_1 +y$ where $x_1 \in L_\Phi (\mathcal {M}_1)$ and y is a scalar multiple of an algebraic $\mathsf {h}_{\Phi }^c$ -atom.

Indeed, if $\Pi : L_\Phi ^{\mathrm {cond}}(\mathcal {M};\ell _2^c) \to \mathsf {h}_\Phi ^c(\mathcal {M})$ denotes the norm one projection described in Lemma 2.3 and $x=\alpha \,. \, \beta $ is a algebraic $L_\Phi ^{c,{ \textrm {cond}}}$ -atom with $\alpha $ being a strictly lower triangular matrix in $L_2(\mathcal {M} \overline {\otimes } B(\ell _2(\mathbb N)))$ and $\beta $ is an adapted column matrix in $L_{\Theta }(\mathcal {M};\ell _2^c)$ , then we have

$$ \begin{align*} \Pi(x) &= \sum_n \sum_{n>k} d_n(\alpha_{n,k}) \beta_{k,1}\\ &=\sum_k \left(\sum_{n>k} d_n(\alpha_{n,k}) \right) \beta_{k,1}\\ &=\sum_k a_k \beta_{k,1}. \end{align*} $$

Clearly, we have for every $ k\geq 1$ , $\mathcal {E}_k(a_k)=0$ . Moreover, $\sum _{k\geq 1} \|a_k\|_2^2 \leq \|\alpha \|_2^2$ . This shows in particular that $\Pi (x)$ is an algebraic $\mathsf {h}_\Phi ^c$ -atom. The assertions in the corollary follow from combining this fact with Theorem 2.8 and direct sum.  ▪

We conclude this section with a companion of Theorem 2.8 for spaces of adapted sequences. It may be viewed as the algebraic atomic decompositions for spaces of adapted sequences. This will be used in the next section.

Proposition 2.12 Let $\Phi $ be a p-convex and q-concave Orlicz function for $0<p<q<2$ . If $x=(x_n)_{n\geq 1}$ is a sequence in $L_\Phi ^{\mathrm {ad}}(\mathcal {M};\ell _2^c)$ then there exists a sequence $\beta =(\beta _n)_{n\geq 1}$ in $L_\Theta ^{\mathrm {ad}}(\mathcal {M};\ell _2^c)$ and a lower triangular matrix $\alpha $ with the following properties:

1. (i) $\alpha \in L_2(\mathcal {M}\overline {\otimes } B(\ell _2(\mathbb N)))$ ;

2. (ii) for every $1\leq j\leq n$ , $\alpha _{n,j} \in L_2(\mathcal {M}_n)$ ;

3. (iii) $x=\alpha \,. \, \beta $ ; and

4. (iv) $\big \|\alpha \big \|_2 \,. \, \big \|\beta \big \|_{L_\Theta (\mathcal {M};\ell _2^c)} \lesssim _{p,q} \big \| x\big \|_{L_\Phi (\mathcal {M};\ell _2^c)}$ .

Conversely, any sequence x admitting a factorization as above belongs to $L_\Phi ^{\mathrm {ad}}(\mathcal {M};\ell _2^c)$ with

$$ \begin{align*} \big\| x\big\|_{L_\Phi(\mathcal{M};\ell_2^c)} \leq \big\|\alpha \big\|_2 \,. \, \big \|\beta \big\|_{L_\Theta(\mathcal{M};\ell_2^c)}. \end{align*} $$

First, we note that since square functions are well-defined for elements of $L_\Phi ^{\mathrm {ad}}(\mathcal {M}; \ell _2^c)$ , reduction to finite sequences or sequences of finite supports is not necessary.

Proof Let $x=(x_n)_{n\geq 1}\in L_\Phi ^{\mathrm {ad}}(\mathcal {M};\ell _2^c)$ . The construction is an adaptation of the proof of Theorem 2.8 but using square functions in place of conditioned square functions.

For each $n\geq 1$ , $\mathcal {S}_{c,n}(x) \in L_{\Phi }(\mathcal {M}_n)$ . That is, $(\mathcal {S}_{c,n}(x))_{n\geq 1}$ is an adapted sequence. We simply write $\varsigma _n$ for $\mathcal {S}_{c,n}(x)$ . As before, we may assume that the $\varsigma _n$ s are invertible with bounded inverse. Similarly, $(\Psi (\varsigma _n))_{n\geq 1}$ is an adapted sequence with the $\Psi (\varsigma _n)$ s being invertible.

As in the proof of Theorem 2.8, fix $\lambda>0$ and set

$$ \begin{align*} \begin{cases} \alpha_{n,1} &:=x_n\Psi(\lambda\varsigma_n)\Psi(\lambda\varsigma_1)^{-1/2};\\ \alpha_{n,j}&:=\displaystyle{x_n\Psi(\lambda\varsigma_n)\left(\Psi(\lambda\varsigma_{j})^{-1}-\Psi(\lambda\varsigma_{j-1})^{-1}\right)^{1/2},\quad\text{for } 2 \leq j \leq n} \end{cases} \end{align*} $$

and

$$ \begin{align*} \begin{cases} \beta_{1,1} &:=\displaystyle{\Psi(\lambda\varsigma_1)^{-1/2};}\\ \beta_{m,1}&:=\displaystyle{\left(\Psi(\lambda\varsigma_{m})^{-1}-\Psi(\lambda\varsigma_{m-1})^{-1}\right)^{1/2}, \quad \text{for }m\geq 2}. \end{cases} \end{align*} $$

A slight difference here is that unlike in the proof of Theorem 2.8, we do not make the indexing shift. The factorization $x=\alpha \, .\, \beta $ is straightforward. Clearly, $(\beta _{m,1})_{m\geq 1}$ is an adapted sequence. Moreover, as $x_n \in L_\Phi (\mathcal {M}_n)$ , it follows that $\alpha _{n,j}$ is affiliated with $\mathcal {M}_n$ and $(\alpha _{n,j})_{1\leq j\leq n}$ is a lower triangular matrix.

We may choose $\lambda $ exactly as in the proof of Theorem 2.8. That is, $\lambda ^{-1}=K_{p,q}^{1/p} \| \mathcal {S}_c(x) \|_{L_\Phi (\mathcal {M})}$ . With this choice, the verification of the fact that $\big \|\alpha \big \|_2\,. \, \big \|\beta \big \|_{L_\Theta (\mathcal {M};\ell _2^c)} \lesssim _{p,q} \big \|x \big \|_{L_\Phi (\mathcal {M};\ell _2^c)}$ follows the same reasoning as in the proof of Theorem 2.8 and is left to the reader.

On the other hand, if x admits a factorization $x=\alpha \, .\, \beta $ , then from the fact that $L_\Phi =L_2 \odot L_\Theta $ , we may conclude as in the proof of Lemma 2.6 that

$$ \begin{align*} \|x \|_{L_\Phi(\mathcal{M};\ell_2^c)} \leq \|\alpha\|_2\,. \, \|\beta\|_{L_\Theta(\mathcal{M};\ell_2^c)}. \end{align*} $$

The proof is complete.  ▪

3.1 Proof of Theorem 3.5

We consider first two important intermediate cases. One is the Banach space range and the other is when the distance between the two indices is small enough. We begin with the latter.

Lemma 3.10 Let $\nu _0 \geq 2$ . Consider $2/(\nu _0+1) <p_0 \leq 2/\nu _0$ and $p_0<p_1 <2/(\nu _0-1)$ . If $0<\theta <1$ , $0<\gamma _0,\gamma _1, \gamma \leq \infty $ , and $1/r=(1-\theta )/p_0 + \theta /p_1$ , then

$$ \begin{align*} \mathsf{h}_{r, \gamma}^c(\mathcal{M})= \big( \mathsf{h}_{p_0,\gamma_0}^c(\mathcal{M}) , \mathsf{h}_{p_1,\gamma_1}^c(\mathcal{M})\big)_{\theta, \gamma} \end{align*} $$

with equivalent quasi-norms.  ▪

Proof This will be deduced from Corollary 3.8 (see also Remark 3.9) and the description of noncommutative Lorentz spaces as real interpolation of the couple $(L_{p_0}(\mathcal {M}), L_{p_1}(\mathcal {M}))$ . Indeed, if $y \in \mathsf {h}_{r,\gamma }^c(\mathcal {M})$ , then we have:

$$ \begin{align*} \big\| y \big\|_{\mathsf{h}_{r,\gamma}^c}&=\big\| U\mathcal{D}_c(y)\big\|_{L_{r,\gamma}(\mathcal{N})}\\ &\approx \big\| U\mathcal{D}_c(y)\big\|_{(L_{p_0}(\mathcal{N}), L_{p_1}(\mathcal{N}))_{\theta, \gamma} }\\ &\approx \big\| y \big\|_{(\mathsf{h}_{p_0}^c(\mathcal{M}),\mathsf{h}_{p_1}^c(\mathcal{M}))_{\theta,\gamma}}, \end{align*} $$

where the last equivalence comes from the comparison of K-functionals stated in Remark 3.9. This clearly shows that $\mathsf {h}_{r, \gamma }^c(\mathcal {M})= \big ( \mathsf {h}_{p_0}^c(\mathcal {M}) , \mathsf {h}_{p_1}^c(\mathcal {M})\big )_{\theta , \gamma }$ . The full generality as stated in the lemma follows by reiteration.  ▪

The next lemma is the infinite version of the Banach space case. It will be deduced from Lemma 3.10 and Wolff’s interpolation theorem. Since our approach differs from [Reference Bekjan, Chen, Perrin and Yin1], we include the details.

Lemma 3.11 Let $1<p<r<\infty $ . If $1/r=(1-\theta )/p$ then

$$ \begin{align*} \mathsf{h}_{r, \gamma}^c(\mathcal{M})= \big( \mathsf{h}_p^c(\mathcal{M}) , \mathsf{bmo}^c(\mathcal{M})\big)_{\theta, r} \end{align*} $$

with equivalent norms.

Proof We will verify first that if $1<r<q<\infty $ then

(3.9) $$ \begin{align} \mathsf{h}_r^c(\mathcal{M})=\big(\mathsf{h}_1^c(\mathcal{M}), \mathsf{h}_q^c(\mathcal{M})\big)_{\psi, r} \end{align} $$

for $1/r=(1-\psi ) +\psi /q$ . We separate the proof of (3.9) into three cases:

$\bullet $ $1<r<q<2$ . This follows immediately from using $\nu _0=2$ , $p_0=1$ , and $p_1=q$ in Lemma 3.10.

$\bullet $ $1<r<2\leq q<\infty $ . Fix v such that $1<r<v<2\leq q$ . Applying the previous case, we have

$$ \begin{align*} \mathsf{h}_r^c(\mathcal{M})=\big(\mathsf{h}_1^c(\mathcal{M}), \mathsf{h}_v^c(\mathcal{M})\big)_{\psi_1, r} \end{align*} $$

for $1/r=(1-\psi _1) +\psi _1/v$ . On the other hand, since for every $1<s<\infty $ , $\mathsf {h}_s^c(\mathcal {M})$ embeds complementably into $L_s(\mathcal {M} \overline {\otimes } B(\ell _2(\mathbb {N}^2)))$ , we have for $1<r<v< q<\infty $ and $1/v= (1-\psi _2)/r + \psi _2/q$ that

$$ \begin{align*} \mathsf{h}_v^c(\mathcal{M})=\big(\mathsf{h}_r^c(\mathcal{M}), \mathsf{h}_q^c(\mathcal{M})\big)_{\psi_2, v}. \end{align*} $$

Set $B_1=\mathsf {h}_1^c(\mathcal {M})$ , $B_2=\mathsf {h}_r^c(\mathcal {M})$ , $B_3=\mathsf {h}_v^c(\mathcal {M})$ , and $B_4=\mathsf {h}_q^c(\mathcal {M})$ . It is clear that $B_1\cap B_4$ is dense in both $B_2$ and $B_3$ . Applying Wolff’s interpolation theorem, we deduce (3.9) with $\psi =\psi _1\psi _2(1-\psi _2 +\psi _1\psi _2)^{-1}$ . One can easily verify that this is the desired index.

$\bullet $ $2\leq r<q<\infty $ . Fix $1<u <2 \leq r<q$ and write

$$ \begin{align*} \mathsf{h}_r^c(\mathcal{M})=\big(\mathsf{h}_u^c(\mathcal{M}), \mathsf{h}_q^c(\mathcal{M})\big)_{\theta_1, r}. \end{align*} $$

Next, we have from the previous case that

$$ \begin{align*} \mathsf{h}_u^c(\mathcal{M})=\big(\mathsf{h}_1^c(\mathcal{M}), \mathsf{h}_q^c(\mathcal{M})\big)_{\theta_2, u}. \end{align*} $$

Applying Wolff’s interpolation theorem with $B_1=\mathsf {h}_1^c(\mathcal {M})$ , $B_3=\mathsf {h}_r^c(\mathcal {M})$ , $B_2=\mathsf {h}_u^c(\mathcal {M})$ , and $B_4= h_q^c(\mathcal {M})$ , the desired interpolation follows with $\psi = \theta _1(1-\theta _2 +\theta _2\theta _1)^{-1}$ . This proves (3.9).

Using the description of the dual of $\mathsf {h}_1^c(\mathcal {M})$ from (2.7) and the well-known fact that $(\mathsf {h}_v^c(\mathcal {M}))^*=\mathsf {h}_{v'}^c(\mathcal {M})$ for $1<v<\infty $ and $v'$ is its conjugate index [Reference Junge and Xu26], we obtain from (3.9) and the duality for interpolation [Reference Bergh and Löfström3, Theorem 3.7.1] that if $1<p<r<\infty $ , then

$$ \begin{align*} \mathsf{h}_r^c(\mathcal{M}) = \big(\mathsf{h}_p^c(\mathcal{M}), \mathsf{bmo}^c(\mathcal{M})\big)_{\theta, r}, \end{align*} $$

which is the desired conclusion.  ▪

We are now ready to present the proof of Theorem 3.5. For $0<p\leq \infty $ , let $A_{p,\gamma } := \mathsf {h}_{p,\gamma }^c(\mathcal {M})$ when $0<p<\infty $ and $A_{\infty , \gamma }:=\mathsf {bmo}^c(\mathcal {M})$ . Consider the sequence of intervals $(I_\nu )_{\nu \geq 1}$ with $I_1=(1,\infty ]$ and for $\nu \geq 2$ ,

$$ \begin{align*} I_\nu= \bigg(\frac{2}{\nu+1}, \frac{2}{\nu-1}\bigg). \end{align*} $$

For a given $ \nu \geq 2$ , it follows from Lemma 3.10 that the family $\{ A_{p,\gamma }\}_{p,\gamma \in (0,\infty ]}$ forms a real-interpolation scale on the interval $I_\nu $ . On the other hand, by reiteration, Lemma 3.11 gives that the family $\{ A_{p,\gamma }\}_{p,\gamma \in (0,\infty ]}$ forms a real-interpolation scale on the interval $I_1$ .

Next, we have $I_1 \cap I_2= (1,2)$ and for $\nu \geq 2$ , $I_\nu \cap I_{\nu +1}= (2/(\nu +1), 2/\nu ]$ . By applying Lemma 3.4 inductively, we deduce that the family $\{ A_{p,\gamma }\}_{p,\gamma \in (0,\infty ]}$ forms a real-interpolation scale on the interval $\bigcup _{\nu =1}^\infty I_\nu = (0,\infty ]$ which is the desired conclusion.  ▪

Adapting the argument above by using Proposition 3.7 in place of Corollary 3.8, we also obtain the corresponding result at the level of conditioned spaces.

Proposition 3.12 If $0<\theta <1$ , $0<p, q<\infty $ , and $0<\gamma \leq \infty $ , then for $1/{r}=(1-\theta )/{p} + {\theta }/q$ ,

$$ \begin{align*} \big( L_p^{\mathrm{cond}}(\mathcal{M}; \ell_2^c), L_q^{\mathrm{cond}}(\mathcal{M}; \ell_2^c)\big)_{\theta, \gamma} = L_{r,\gamma}^{\mathrm{cond}}(\mathcal{M};\ell_2^c) \end{align*} $$

with equivalent quasi-norms.

3.2 Spaces of adapted sequences

In this subsection, we apply ideas used for the case of conditioned Hardy spaces to the family of spaces of adapted sequences. We first observe that by the noncommutative Stein inequality [Reference Pisier and Xu37], the space of adapted sequences $L_p^{\mathrm {ad}}(\mathcal {M};\ell _2^c)$ is complemented in $L_p(\mathcal {M};\ell _2^c)$ when $1<p<\infty $ . Thus, it is rather an easy task to see that the family $\{ L_p^{\mathrm {ad}}(\mathcal {M};\ell _2^c)\}_{1<p<\infty }$ forms interpolation scales. The next result extends this fact to the full range $0<p<\infty $ .

Theorem 3.13 If $0<\theta <1$ , $0<p, q<\infty $ , and $0<\gamma \leq \infty $ , then for $1/{r}=(1-\theta )/{p} + {\theta }/q$ ,

$$ \begin{align*} \big( L_p^{\mathrm{ad}}(\mathcal{M}; \ell_2^c), L_q^{\mathrm{ad}}(\mathcal{M}; \ell_2^c)\big)_{\theta, \gamma} = L_{r,\gamma}^{\mathrm{ad}}(\mathcal{M};\ell_2^c) \end{align*} $$

with equivalent quasi-norms.

As in the case of conditioned spaces, the proof is based on estimates of K-functionals. We observe first that since $L_r(\mathcal {M};\ell _2^c)$ is a one-complemented subspace of $L_r(\mathcal {M} \overline {\otimes } B(\ell _2))$ for every $0<r<\infty $ , one can easily compute the K-functionals for the couple $(L_p(\mathcal {M};\ell _2^c),L_q(\mathcal {M};\ell _2^c))$ . Adapting the argument used in the proof of Proposition 3.7 (using Proposition 2.12 in place of Theorem 2.8 in the first step), we obtain the corresponding result for spaces of adapted sequences. More precisely:

Proposition 3.14 Let $\nu $ be a positive integer with $\nu \geq 2$ . Assume that $2/(\nu +1) <p\leq 2/\nu $ and $p <q <2/(\nu -1)$ . For every sequence $a \in L_p^{\mathrm {ad}}(\mathcal {M};\ell _2^c)+ L_q^{\mathrm {ad}}(\mathcal {M};\ell _2^c)$ and every $t>0$ , the following holds:

$$ \begin{align*} K\big(a,t; L_p^{\mathrm{ad}}(\mathcal{M};\ell_2^c), L_q^{\mathrm{ad}}(\mathcal{M};\ell_2^c)\big) \approx_{p,q} K\big( \mathcal{S}_c(a), t; L_p(\mathcal{M}), L_q(\mathcal{M})\big). \end{align*} $$

Proof (Sketch of the proof of Theorem 3.13)

First, we use Proposition 3.14 to deduce the corresponding result to Lemma 3.10. Fix $\nu _0 \geq 2$ . Assume that $2/(\nu _0+1) <p_0 \leq 2/\nu _0$ and $p_0<p_1 <2/(\nu _0-1)$ . If $0<\theta <1$ , $0<\gamma _0,\gamma _1, \gamma \leq \infty $ , and $1/r=(1-\theta )/p_0 + \theta /p_1$ , then

(3.10) $$ \begin{align} L_{r, \gamma}^{\mathrm{ad}}(\mathcal{M};\ell_2^c)= \big( L_{p_0,\gamma_0}^{\mathrm{ad}}(\mathcal{M};\ell_2^c) , L_{p_1,\gamma_1}^{\mathrm{ad}}(\mathcal{M}; \ell_2^c)\big)_{\theta, \gamma}. \end{align} $$

Next, we deduce the Banach space range using complementation: if $1< p<r <q<\infty $ and $1/r=(1-\psi )/p + \psi /q$ , then

(3.11) $$ \begin{align} L_r^{\mathrm{ad}}(\mathcal{M};\ell_2^c) =\big( L_{p}^{\mathrm{ad}}(\mathcal{M};\ell_2^c) , L_{q}^{\mathrm{ad}}(\mathcal{M}:\ell_2^c)\big)_{\psi, r}. \end{align} $$

Using (3.10) and (3.11), we can repeat the inductive argument used in the proof of Theorem 3.5 with the intervals $I_1=(1,\infty )$ and $I_\nu =({2}/(\nu +1), {2}/(\nu -1))$ for $\nu \geq 2$ , to conclude that the family $\{L_p^{\mathrm {ad}}(\mathcal {M};\ell _2^c)\}_{p\in (0, \infty ); \gamma \in (0,\infty ]}$ forms a real interpolation scale.  ▪

Assume that $1\leq p<\infty $ . We recall that the map $\mathcal {D}(x)= (dx_n)_{n\geq 1}$ is an isometric embedding of $\mathcal {H}_p^c(\mathcal {M})$ into $L_p^{\mathrm {ad}}(\mathcal {M};\ell _2^c)$ . Using the noncommutative Stein inequality when $1<p<\infty $ and the noncommutative Lépingle-Yor inequality when $p=1$ [Reference Qiu39], the linear map

$$ \begin{align*} \Pi \big( (a_n)_{n\geq 1}\big)= (a_n-\mathcal{E}_{n-1}(a_n))_{n\geq 1} \end{align*} $$

is simultaneously bounded from $L_p^{\mathrm {ad}}(\mathcal {M};\ell _2^c)$ onto $\mathcal {D}\big (\mathcal {H}_p^c(\mathcal {M})\big )$ for all $1\leq p<\infty $ . As a result, one can immediately deduce from Theorem 3.13 that the interpolation

$$ \begin{align*}(\mathcal{H}_1^c(\mathcal{M}), \mathcal{H}_q^c(\mathcal{M}))_{\theta, r} = \mathcal{H}_r^c(\mathcal{M}) \end{align*} $$

for $0<\theta <1$ , $1<q<\infty $ , and $ 1/r=(1-\theta ) +\theta /q$ holds. Due to this fact, we may view Theorem 3.13 as an extension of the real interpolation version of [Reference Musat32] to the quasi-Banach space range. However, when $0<p<1$ , the spaces of adapted sequences cannot be replaced by column martingale Hardy spaces. In fact, only one inclusion holds if column martingale Hardy spaces are used. The next result may be viewed as a noncommutative generalization of [Reference Janson and Jones18, Theorem 2]. We also direct the reader to its companion Corollary 3.19 below and a discussion on the non validity of the reverse inclusions. We refer to [Reference Musat32, Reference Pisier and Xu37] for the definition of ${\mathcal {BMO}}^c(\mathcal {M})$ .

Corollary 3.15 Assume that $0<p<1$ and $0<\theta <\infty $ are such that $1/r=(1-\theta )/p <1$ . Then

$$ \begin{align*} \big( \mathcal{H}_p^c(\mathcal{M}), {\mathcal {BMO}}^c(\mathcal{M}) \big)_{\theta, r}\subset \mathcal{H}_r^c(\mathcal{M}). \end{align*} $$

Proof Let $0<\eta <1$ such that $1= (1-\eta )/p +\eta /r$ . Let $ x\in \big (\mathcal {H}_p^c(\mathcal {M}), \mathcal {H}_r^c(\mathcal {M}) \big )_{\eta , 1}$ . Since $\mathcal {H}_u^c(\mathcal {M}) \subset L_u^{\mathrm {ad}}(\mathcal {M};\ell _2^c)$ for $u\in \{p,r\}$ , we have

$$ \begin{align*} \|x\|_{\mathcal{H}_1^c} &=\| dx\|_{L_1^{\mathrm{ad}}(\mathcal{M};\ell_2^c)}\\ &\approx \| dx \|_{(L_p^{\mathrm{ad}}(\mathcal{M};\ell_2^c), L_r^{\mathrm{ad}}(\mathcal{M};\ell_2^c))_{\eta, 1}}\\ &\leq \|x \|_{(\mathcal{H}_p^c(\mathcal{M}), \mathcal{H}_r^c(\mathcal{M}))_{\eta, 1}}, \end{align*} $$

where the second equivalence comes from Proposition 3.13. This shows that

$$ \begin{align*} \big( \mathcal{H}_p^c(\mathcal{M}), \mathcal{H}_r^c(\mathcal{M}) \big)_{\eta, 1}\subset \mathcal{H}_1^c(\mathcal{M}). \end{align*} $$

Next, we recall that $\big (\mathcal {H}_1^c(\mathcal {M}), {\mathcal {BMO}}^c(\mathcal {M})\big )_{\phi , r} =\mathcal {H}_r^c(\mathcal {M})$ where $1/r=1-\phi $ . We remark that the Wolff’s interpolation theorem is valid at the level of inclusion: if $B_1 \cap B_4 \subset B_2 \cap B_3$ , $(B_1, B_3)_{\eta , q_1} \subset B_2$ , and $(B_2, B_4)_{\phi , q_2} \subset B_3$ , then $(B_1, B_4)_{\theta , q_3} \subset B_3$ whenever $0<q_j\leq \infty $ ( $j=1,2,3$ ) and $\theta =\phi /(1-\eta +\eta \phi )$ . The verification of this fact can be found in the first part of the proof of [Reference Wolff48, Theorem 1]. Using $B_1=\mathcal {H}_p^c(\mathcal {M})$ , $B_2=\mathcal {H}_1^c(\mathcal {M})$ , $B_3= \mathcal {H}_r^c(\mathcal {M})$ , and $B_4={\mathcal {BMO}}^c(\mathcal {M})$ , we obtain the desired conclusion.  ▪

3.3 The complex method

We now turn our attention to the case of complex interpolation method which we now briefly review.

Let S (respectively, $\overline {S}$ ) denote the open strip $\{z : 0< \operatorname {Re} z<1\}$ (respectively, the closed strip $\{z : 0\leq \operatorname {Re} z \leq 1\}$ ) in the complex plane $\mathbb C$ . Let $A(S)$ be the collection of $\mathbb C$ -valued functions that are analytic on S and continuous and bounded on $\overline {S}$ . For a compatible couple of complex quasi-Banach spaces $(A_0,A_1)$ , we denote by $\mathcal {F}_0(A_0,A_1)$ the family of functions of the form $f(z)= \sum _{k=1}^n f_k(z)x_k$ with $f_k \in A(S)$ and $x_k \in A_0 \cap A_1$ . We equip $\mathcal {F}_0(A_0,A_1)$ with the quasi-norm:

$$ \begin{align*} \big\|f\big\|_{\mathcal{F}_0(A_0,A_1)}= \max \Big\{ \mathop{\mathrm{sup}}_{t\in \mathbb R} \big\| f(it)\big\|_{A_0} , \mathop{\mathrm{sup}}_{t\in \mathbb R} \big\| f(1+it)\big\|_{A_1}\Big\}. \end{align*} $$

Then $\mathcal {F}_0(A_0,A_1)$ becomes a quasi-Banach space. For $0<\theta <1$ , the complex interpolation norm on $A_0 \cap A_1$ is defined by:

$$ \begin{align*} \big\|x \big\|_{[A_0,A_1]_\theta} =\inf\Big\{\big\|f\big\|_{\mathcal{F}_0(A_0,A_1)} : f(\theta)=x,\ f \in \mathcal{F}_0(A_0,A_1)\Big\}. \end{align*} $$

The complex interpolation space (of exponent $\theta $ ) $[A_0,A_1]_\theta $ is defined as the completion of the quasi-normed space $(A_0 \cap A_1, \| \cdot \|_{[A_0,A_1]_\theta })$ .

As in the real method, complex interpolations of the couple $(L_p, L_q)$ (for $0<p<q\leq \infty $ ) are well-known. Indeed, if $\mathcal {N}$ is a semifinite von Neumann algebra and $0<\theta <1$ , then

$$ \begin{align*} \big[L_p(\mathcal{N}), L_q(\mathcal{N})\big]_\theta =L_r(\mathcal{N}) \end{align*} $$

isometrically for $1/r=(1-\theta )/p +\theta /q$ . This fact comes from [Reference Xu49, Theorem 4.1] (see also [Reference Pisier and Xu38]).

The next result is the version of Theorem 3.5 for the complex method.

Theorem 3.16 If $0<p,q<\infty $ , $0<\theta <1$ , and $ {\frac {1}{r}=\frac {1-\theta }{p} + \frac {\theta }{q}}$ , then

$$ \begin{align*} \mathsf{h}_r^c(\mathcal{M}) =\big[\mathsf{h}_p^c(\mathcal{M}), \mathsf{h}_q^c(\mathcal{M}) \big]_\theta \end{align*} $$

with equivalent quasi-norms.

For the proof, we will use the next result which provides a connection between complex interpolation method and real interpolation method that is valid for quasi-Banach spaces. We should note that for Banach spaces, the inequality in the next theorem is actually an equivalence but for quasi-Banach spaces only one inequality is valid in its full generality.

Theorem 3.17 [Reference Cwikel, Milman and Sagher9, Theorem 3]

Let $(A_0, A_1)$ be a compatible couple of quasi-Banach spaces. Let $0<\theta _j<1$ , $0<\theta <1$ , and $0<\gamma _j\leq \infty $ . Denote $E_j= (A_0, A_1)_{\theta _j, \gamma _j}$ for $j=0,1$ . If $1/\gamma = (1-\theta )/\gamma _0 +\theta /\gamma _1$ and $\lambda =(1-\theta )\theta _0 +\theta \theta _1$ then for every $a \in A_0\cap A_1\subset E_0 \cap E_1$ ,

$$ \begin{align*} \big\| a \big\|_{[E_0,E_1]_\theta} \leq C \big\|a \big\|_{(A_0, A_1)_{\lambda, \gamma}}. \end{align*} $$

Proof (of Theorem 3.16)

Assume that $0<p<q<\infty $ and fix $0<p_0<p$ . Then according to Theorem 3.5, $\mathsf {h}_p^c(\mathcal {M})=(\mathsf {h}_{p_0}^c(\mathcal {M}), \mathsf {bmo}^c(\mathcal {M}))_{\theta _0, p}$ and $h_q^c(\mathcal {M})= (\mathsf {h}_{p_0}^c(\mathcal {M}), \mathsf {bmo}^c(\mathcal {M}))_{\theta _1, q}$ where $1/p=(1-\theta _0)/p_0$ and $1/q=(1-\theta _1)/p_0$ . We may state from Theorem 3.17 that if $1/\gamma = (1-\theta )/p +\theta /q$ and $\lambda =(1-\theta )\theta _0 +\theta \theta _1$ then for every $a\in \mathsf {h}_{p_0}^c(\mathcal {M}) \cap \mathsf {bmo}^c(\mathcal {M})$ ,

$$ \begin{align*} \big\| a \big\|_{[\mathsf{h}_p^c(\mathcal{M}), \mathsf{h}_q^c(\mathcal{M})]_\theta} \leq C \big\|a\big\|_{(\mathsf{h}_{p_0}^c(\mathcal{M}), \mathsf{bmo}^c(\mathcal{M}))_{\lambda, \gamma}}. \end{align*} $$

One can easily verify that $(1-\lambda )/{p_0}=1/\gamma =1/r$ and therefore we may deduce from Theorem 3.5 that

(3.12) $$ \begin{align} \big\| a \big\|_{[\mathsf{h}_p^c(\mathcal{M}), \mathsf{h}_q^c(\mathcal{M})]_\theta} \leq C' \big\|a\big\|_{\mathsf{h}_r^c}. \end{align} $$

On the other hand, let $U: L_s^{\mathrm {cond}}(\mathcal {M};\ell _2^c) \to L_s(\mathcal N)$ (where $\mathcal {N}=\mathcal {M}\overline {\otimes } B(\ell _2(\mathbb {N}^2))$ ) be the family of isometric embeddings as described in the preliminary section which are valid for all $0<s\leq \infty $ . Denote by $\mathcal {D}_c$ the extension of the map $x\mapsto (dx_n)_{n\geq 1}$ from $\mathsf {h}_s^c(\mathcal {M})$ into $L_s^{\mathrm {cond}}(\mathcal {M};\ell _2^c)$ . Then for every $0<s\leq \infty $ , $U\mathcal {D}_c$ is an isometric embedding of $\mathsf {h}_s^c(\mathcal {M})$ into $L_s(\mathcal {N})$ . Let $b \in \mathsf {h}_p^c(\mathcal {M}) \cap \mathsf {h}_q^c(\mathcal {M})$ . Interpolating the operator $U\mathcal {D}_c$ , we have

$$ \begin{align*} \big\| U\mathcal{D}_c(b)\big\|_{[L_p(\mathcal{N}), L_q(\mathcal{N})]_\theta} \leq \big\| b \big\|_{[\mathsf{h}_p^c(\mathcal{M}), \mathsf{h}_q^c(\mathcal{M})]_\theta}. \end{align*} $$

Since $\big [(L_p(\mathcal {N}), L_q(\mathcal {N})\big ]_\theta =L_r(\mathcal {N})$ isometrically, it follows that

$$ \begin{align*} \big\| U\mathcal{D}_c(b)\big\|_{L_r(\mathcal{N})} \leq \big\| b \big\|_{[\mathsf{h}_p^c(\mathcal{M}), \mathsf{h}_q^c(\mathcal{M})]_\theta}. \end{align*} $$

From the fact that $U\mathcal {D}_c$ is an isometry on $\mathsf {h}_r^c(\mathcal {M})$ , we deduce that

(3.13) $$ \begin{align} \big\|b\big\|_{\mathsf{h}_r^c} \leq \big\| b \big\|_{[\mathsf{h}_p^c(\mathcal{M}), \mathsf{h}_q^c(\mathcal{M})]_\theta}. \end{align} $$

From combining (3.12) and (3.13), we obtain the desired equivalence.  ▪

When $0<p<1$ , we do not know if the corresponding statement to Theorem 3.16 remains valid if the interpolation couple $(\mathsf {h}_p^c(\mathcal {M}), \mathsf {bmo}^c(\mathcal {M}))$ is used. Since reiteration theorem is not available for complex interpolations of quasi-Banach spaces, in general, this consideration is independent of Theorem 3.16. We leave this as an open problem.

The same method of proofs can be applied to conditioned spaces and spaces of adapted sequences to deduce the following interpolation results from Proposition 3.12 and Theorem 3.13, respectively.

Proposition 3.18 If $0<\theta <1$ and $0<p, q<\infty $ then for $1/{r}=(1-\theta )/{p} + {\theta }/q$ , the following hold:

$$ \begin{align*} \big[ L_p^{\mathrm{cond}}(\mathcal{M}; \ell_2^c), L_q^{\mathrm{cond}}(\mathcal{M}; \ell_2^c)\big]_{\theta} = L_{r}^{\mathrm{cond}}(\mathcal{M};\ell_2^c) \end{align*} $$

and

$$ \begin{align*} \big[ L_p^{\mathrm{ad}}(\mathcal{M}; \ell_2^c), L_q^{\mathrm{ad}}(\mathcal{M}; \ell_2^c)\big]_{\theta} = L_{r}^{\mathrm{ad}}(\mathcal{M};\ell_2^c) \end{align*} $$

with equivalent quasi-norms.

As in the case of real interpolation, we may view the second assertion in the proposition as an extension of Musat’s result to the quasi-Banach space range. Moreover, using [Reference Wolff48, Lemma 1] (which is valid for quasi-Banach spaces), one can adapt the argument used in the proof of Corollary 3.15 to show that the noncommutative analogue of (1.2) holds:

Corollary 3.19 Assume that $0<p<1$ and $0<\theta <\infty $ are such that $1/r=(1-\theta )/p <1$ . Then

$$ \begin{align*} \big[ \mathcal{H}_p^c(\mathcal{M}), {\mathcal {BMO}}^c(\mathcal{M}) \big]_{\theta}\subset \mathcal{H}_r^c(\mathcal{M}). \end{align*} $$

Our method of proof only applies under the assumption that $r>1$ . We should point out that the reverse inclusion does not hold even in the classical setting. An example exhibited in [Reference Janson and Jones18, p. 66] shows that there exists a probability space $(\Omega , \mathcal {F},\mathbb {P})$ and an increasing filtration of $\sigma $ -fields $(\mathcal {F}_n)_{n\geq 1}$ of $\mathcal {F}$ with $\mathcal {F}=\sigma (\bigcup _{n\geq 1} \mathcal {F}_n)$ and such that if $0<p<1<r<\infty $ , then for $1/r=(1-\theta )/p$ , $\big [ \mathcal {H}_p(\Omega ), {\mathcal {BMO}}(\Omega ) \big ]_{\theta } \neq \mathcal {H}_r(\Omega )$ .

All results stated in this section have row counterparts. However, at the time of this writing, it is unclear if for $0<p,q<1$ , the interpolation results for column/row conditioned Hardy spaces have counterparts to the couple of diagonal Hardy spaces $(\mathsf {h}_p^d(\mathcal {M}),\mathsf {h}_q^d(\mathcal {M}))$ .