The one-to-one correspondence between one-dimensional linear (stationary, causal) input/state/output systems and scattering systems with one evolution operator, in which the scattering function of the scattering system coincides with the transfer function of the linear system, is well understood, and has significant applications in H ∞ control theory. Here we consider this correspondence in the d-dimensional setting in which the transfer and scattering functions are defined on the polydisk. Unlike in the onedimensional case, the multidimensional state space realizations and the corresponding multi-evolution scattering systems are not necessarily equivalent, and the cases d = 2 and d > 2 differ substantially. A new proof of Andô's dilation theorem for a pair of commuting contraction operators and a new statespace realization theorem for a matrix-valued inner function on the bidisk are obtained as corollaries of the analysis. (2000). Primary 47A20; Secondary 47A40, 93C55, 93C35. Mathematics Subject Classification

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Abstract. 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 ∞ (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.