CPD $n$th roots of subnormal operators are subnormal

Abstract

We investigate the $n$th root problem for bounded operators on a Hilbert space within the class of conditionally positive definite (CPD) operators determined by the L\'evy--Khintchine formula. The 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 (resp., quasinormal, normal, or a $3$-isometry), then $T$ belongs to the corresponding class. This establishes the invariance of these classes under taking $n$th roots within the CPD class and extends several earlier results in operator theory. Furthermore, we provide characterizations of quasinormal and normal operators in terms of their CPD property and the structure of the representing triplet. Finally, we show that the classes of CPD and normaloid operators are distinct by means of both theoretical arguments and explicit examples.