Within the class of CPD operators, nth roots preserve subnormality, quasinormality, normality, and the 3-isometry property.
Schm¨ udgen, Unbounded self-adjoint operators on Hilbert space, Graduate Texts in Math- ematics
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CPD $n$th roots of subnormal operators are subnormal
Within the class of CPD operators, nth roots preserve subnormality, quasinormality, normality, and the 3-isometry property.