An operator has a Parseval frame of iterations exactly when it is a contraction with strongly stable adjoint, and the minimal generator count equals the dimension of the range of I minus TT*.
Helson.Lectures on Invariant Subspaces.Academic Press, London, 1964
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Optimal Dynamical Frames
An operator has a Parseval frame of iterations exactly when it is a contraction with strongly stable adjoint, and the minimal generator count equals the dimension of the range of I minus TT*.