CPD nth Roots of Subnormal Operators are Subnormal

Abstract

We investigate the nth-root problem for bounded operators on a Hilbert space within the class of conditionally positive definite (CPD) operators determined by the Lévy–Khintchine formula. This class contains subnormal operators, complete hypercontractions of order 2, and 3-isometries. Our main result shows that if T is a CPD operator such that $$T^n$$ is subnormal (respectively, quasinormal, normal, or a 3-isometry), then T belongs to the corresponding class. This establishes that these classes are invariant under taking nth roots within the CPD class and extends several earlier results in operator theory. Furthermore, we characterize quasinormal and normal operators in terms of the CPD property and the structure of the associated representing triplet. Finally, using both theoretical arguments and explicit examples, we show that the classes of CPD and normaloid operators are distinct.