The Nehari problem for the Hardy space on the torus

We explicitly construct functions in H-2(T-2)(perpendicular to) which deter mine bounded (big) Hankel operators on H-2(T-2) but are not of the form P(p erpendicular to)psi for any psi is an element of L-infinity(T-2). We use th is construction to show that the norm of a Hankel operator with bounded sym bol is not, in general, comparable to the distance the symbol is from H-inf inity(T-2). We also characterize the vector space quotient of symbols of bo unded Hankel operators module those which lift to L-infinity(T-2) in terms of a Toeplitz completion problem on vector-valued Hardy space in one-variab le.