Two Distinguished Subspaces of Product BMO and the Nehari--AAK Theory for Hankel Operators on the Torus

In this paper we show that the theory of Hankel operators in the torus $\T^d$, for $d > 1$, presents striking differences with that on the circle $\T$, starting with bounded Hankel operators with no bounded symbols. Such differences are circumvented here by replacing the space of symbols $L^\infty (\T)$ by BMOr$(\T^d)$, a subspace of product BMO, and the singular numbers of Hankel operators by so-called sigma numbers. This leads to versions of the Nehari--AAK and Kronecker theorems, and provides conditions for the existence of solutions of product Pick problems through finite Pick-type matrices. We give geometric and duality characterizations of BMOr, and of a subspace of it, bmo, closely linked with $A_2$ weights. This completes some aspects of the theory of BMO in product spaces.