Invariant subspaces and extremum problems in spaces of vector-valued functions

In this article we obtain, for 1 ? p ? ∞, a characterization of the invariant subspaces of spaces of vector-valued L^p functions defined on the unit circle—i.e., of those subspaces invariant under multiplication by e^ix. This result is then applied to extend, to the corresponding Hardy classes of vector-valued functions, the known characterizations of the extreme points of the unit ball in the scalar Hardy classes H¹ and H^∞. Finally, it is shown that the characterization of the closure of the set of extreme points of the unit ball in H¹ changes significantly when we pass from the scalar to the vector case.