Quadratically and cubically hyponormal matrix-valued weighted shifts with two equal weights are flat, and 2-hyponormal ones satisfy local flatness yielding a structural decomposition theorem.
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Propagation Phenomena for Operator-Valued Weighted Shifts
Quadratically and cubically hyponormal matrix-valued weighted shifts with two equal weights are flat, and 2-hyponormal ones satisfy local flatness yielding a structural decomposition theorem.