Two Distinguished Subspaces of Product BMO and the Nehari–AAK Theory for Hankel Operators on the Torus
Mischa Cotlar and Cora Sadosky
In this paper we show that the theory of Hankel operators in the torus ${\mathbb T}^d$, for $d \gt 1$, presents striking differences with that on the circle ${\mathbb T}$, starting with bounded Hankel operators with no bounded symbols. Such differences are circumvented here by replacing the space of symbols $L^\infty ({\mathbb T})$ by BMOr$({\mathbb T}^d)$, 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.