On module maps and bilinear forms defined on shift-invariant subspaces of \(L^ p(X)\) (Q1187986)

A lifting of the commutant theorem on \(L^ p(\mathbb{T};X)\) is proved when \(X\) is a Banach space of type 2 and \(p\in [2,+\infty[\). A generalization of Sarason's lifting theorem is obtained as well. Then some modular versions of Hilbertian factorizations for module maps between invariant subspaces of \(L^ p(\mathbb{T};X)\) are studied. The last part of the paper is devoted to a representation theorem for invariant subspaces of \(H^ 2(X)\) which are isomorphic to a Hilbert space.