Lax-Halmos Type Theorems in H^p Spaces

TL;DR

This paper extends the Lax-Halmos theorem to characterize invariant subspaces of Hp spaces under specific powers of a finite Blaschke factor, generalizing previous results and applicable to constrained interpolation problems.

Contribution

It provides new characterizations of invariant subspaces in Hp spaces for certain powers of Blaschke factors, generalizing prior invariant subspace theorems.

Findings

Characterization of subspaces invariant under powers of a finite Blaschke factor.

Generalization of the invariant subspace theorem by Paulsen and Singh.

Method adaptable to invariance under all positive powers of B(z).

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.