A version of the Grothendieck theorem for subspaces of analytic functions in lattices

A version of Grothendieck's inequality says that any bounded linear operator acting from a Banach lattice $X$ to a Banach lattice $Y$, also acts from $X(\ell^2)$ to $Y(\ell^2)$. A similar statement is proved for Hardy-type subspaces in lattices of measurable functions. Namely, let $X$ be a Banach lattice of measurable functions on the circle, and let an operator $T$ act from the corresponding subspace of analytic functions $X_A$ to a Banach lattice $Y$ or, if $Y$ is also a lattice of measurable functions on the circle, to the quotient space $Y/Y_A$. Under certain mild conditions on the lattices involved, it is proved that $T$ induces an operator acting from $X_A(\ell^2)$ to $Y(\ell^2)$ or to $Y/Y_A(\ell^2)$, respectively.