  A Haar system Hardy space is the completion of the linear span of the Haar
system $(h_I)_I$, either under a rearrangement-invariant norm $\|\cdot \|$ or
under the associated square function norm \begin{equation*}
  \Bigl\| \sum_Ia_Ih_I \Bigr\|_{*} = \Bigl\| \Bigl( \sum_I a_I^2 h_I^2
\Bigr)^{1/2} \Bigr\|. \end{equation*} Apart from $L^p$, $1\le p<\infty$, the
class of these spaces includes all separable rearrangement-invariant function
spaces on $[0,1]$ and also the dyadic Hardy space $H^1$. Using a unified and
systematic approach, we prove that a Haar system Hardy space $Y$ with $Y\ne
C(\Delta)$ ($C(\Delta)$ denotes the continuous functions on the Cantor set) has
the following properties, which are closely related to the primariness of $Y$:
For every bounded linear operator $T$ on $Y$, the identity $I_Y$ factors either
through $T$ or through $I_Y - T$, and if $T$ has large diagonal with respect to
the Haar system, then the identity factors through $T$. In particular, we
obtain that \begin{equation*}
  \mathcal{M}_Y = \{ T\in \mathcal{B}(Y) : I_Y \ne ATB\text{ for all } A, B\in
\mathcal{B}(Y) \} \end{equation*} is the unique maximal ideal of the algebra
$\mathcal{B}(Y)$ of bounded linear operators on $Y$. Moreover, we prove similar
factorization results for the spaces $\ell^p(Y)$, $1\le p \leq \infty$, and use
them to show that they are primary.
