Rank of a co-doubly commuting submodule is 2
Proceedings of the American Mathematical Society
We prove that the rank of a non-trivial co-doubly commuting submodule is 2. More precisely, let φ, ψ ∈ H∞(ð») be two inner functions. If (Formula Presented), then rank (Qφ ⊗Qψ)⊥ = 2. An immediate consequence is the following: Let S be a co-doubly commuting submodule of H2 (ð»2). Then rank S = 1 if and only if S = ΦH2 (ð»2) for some one variable inner function Φ ∈ H∞ (ð»2). This answers a question posed by R. G. Douglas and R. Yang [Integral Equations Operator Theory 38(2000), pp207–221].
Chattopadhyay, Arup; Das, B. Krishna; and Sarkar, Jaydeb, "Rank of a co-doubly commuting submodule is 2" (2018). Journal Articles. 1598.