Rank of a co-doubly commuting submodule is 2

Abstract

We prove that the rank of a non-trivial co-doubly commuting sub-module is 2. More precisely, let phi,psi is an element of H-infinity(D) be two inner functions. If Q(phi) = H-2(D)/phi H-2(D) and Q(psi) = H-2(D)/psi H-2(D), then rank (Q(phi) circle times Q(psi))(perpendicular to) = 2. An immediate consequence is the following: Let S be a co-doubly commuting submodule of H-2(D-2). Then rank S = 1 if and only if S = Phi H-2(D-2) for some one variable inner function Phi is an element of H-infinity(D-2). This answers a question posed by R. G. Douglas and R. Yang [ Integral Equations Operator Theory 38(2000), pp207-221]