Rank of a co-doubly commuting submodule is 2

Article Type

Research Article

Publication Title

Proceedings of the American Mathematical Society

Abstract

We prove that the rank of a non-trivial co-doubly commuting submodule is 2. More precisely, let φ, ψ ∈ H∞(𝔻) 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 (𝔻2). Then rank S = 1 if and only if S = ΦH2 (𝔻2) for some one variable inner function Φ ∈ H∞ (𝔻2). This answers a question posed by R. G. Douglas and R. Yang [Integral Equations Operator Theory 38(2000), pp207–221].

First Page

1181

Last Page

1187

DOI

10.1090/proc/13792

Publication Date

1-1-2018

Comments

All Open Access, Bronze, Green

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