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

D. S. Anisimov

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 33, Tome 327 (2005), pp. 5-16

Résumé

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.