SP and SRP are equivalent for recursively generated shifts iff a Mellin transform functional equation holds, with equivalence for p≤6 atomic measures but failure for a specific 7-atomic measure.
Conway, The Theory of Subnormal Operators , Mathematical Surveys and Monographs, vol
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The Square Root Problem and Subnormal Aluthge Transforms of Recursively Generated Weighted Shifts
SP and SRP are equivalent for recursively generated shifts iff a Mellin transform functional equation holds, with equivalence for p≤6 atomic measures but failure for a specific 7-atomic measure.