Two distinguished subspaces of product BMO and Nehari-AAK theory for Hankel operators on the torus

In this paper we show that the theory of Hankel operators in the torus^d, ford>1, presents striking differences with that on the circle, starting with bounded Hankel operators with no bounded symbols. Such differences are circumvented here by replacing the space of symbolsL() by BMOr(^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 Picktype matrices. We give geometric and duality characterizations of BMOr, and of a subspace of it, bmo, closely linked withA₂ weights. This completes some aspects of the theory of BMO in product spaces.Sadosky was partially supported by NSF grants DMS-9205926, INT-9204043 and GER-9550373, and her visit to MSRI is supported by NSF grant DMS-9022140 to MSRI.