Lax-Halmos Type Theorems in H^p Spaces

Abstract: In this paper we characterize for 0 < p \leq \infty, the closed subspaces of Hp that are invariant under multiplication by all powers of a finite Blaschke factor B, except the first power. Our result clearly generalizes the invariant subspace theorem obtained by Paulsen and Singh [9] which has proved to be the starting point of important work on constrained Nevanlinna-Pick interpolation. Our method of proof can also be readily adapted to the case where the subspace is invariant under all positive powers of B (z). The two results are in the mould of the classical Lax-Halmos Theorem and can be said to be Lax-Halmos type results in the finitre multiplicity case for two commuting shifts and for a single shift respectively.