A pair of commuting operators on a separable Hilbert space generates an overcomplete frame by unilateral iterations if and only if it is similar to the two-variable Jordan block on an infinite-dimensional quotient module of the bidisc Hardy space.
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Submodules of $H^2(\mathbb{T}^2)$ and Frames by Pairs of Bounded Commuting Operators
A pair of commuting operators on a separable Hilbert space generates an overcomplete frame by unilateral iterations if and only if it is similar to the two-variable Jordan block on an infinite-dimensional quotient module of the bidisc Hardy space.