CPD $n$th roots of subnormal operators are subnormal
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arXiv
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| Format: | Preprint |
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2026
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| author | Jabłoński, Zenon Jan Jung, Il Bong Pietrzycki, Paweł Stochel, Jan |
| author_facet | Jabłoński, Zenon Jan Jung, Il Bong Pietrzycki, Paweł Stochel, Jan |
| contents | 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 Lévy--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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_10810 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | CPD $n$th roots of subnormal operators are subnormal Jabłoński, Zenon Jan Jung, Il Bong Pietrzycki, Paweł Stochel, Jan Functional Analysis Primary 47B20, 47B15, Secondary 44A60, 43A35 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 Lévy--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. |
| title | CPD $n$th roots of subnormal operators are subnormal |
| topic | Functional Analysis Primary 47B20, 47B15, Secondary 44A60, 43A35 |
| url | https://arxiv.org/abs/2604.10810 |