Geodesic
A version of the Grothendieck theorem for subspaces of analytic functions in lattices
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 33, Tome 327 (2005), pp. 5-16 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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D. S. Anisimov. A version of the Grothendieck theorem for subspaces of analytic functions in lattices. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 33, Tome 327 (2005), pp. 5-16. http://geodesic.mathdoc.fr/item/ZNSL_2005_327_a0/

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