Subnormal $n$th roots of quasinormal operators are quasinormal

TL;DR

The paper resolves the question of whether a subnormal operator $A$ with $A^n$ quasinormal must be quasinormal, proving an affirmative answer for all $n>1$. It develops two independent proofs: one leveraging operator-monotone functions and Hansen's inequality via Embry's characterization, and another using the Stieltjes moment problem together with measure transport to obtain a spectral measure and quasinormality. A generalization is presented for the identity $T^{* abla}T^{ abla}=(T^*T)^{ abla}$ with fixed $ abla>1$, showing subnormal roots preserve quasinormality, along with a spectrality criterion for semispectral measures. These results deepen the link between subnormal and quasinormal operator theory and provide new tools for analyzing roots and moment problems in operator classes.

Abstract

In the recent paper, R. E. Curto, S. H. Lee, J. Yoon, asked the following question: Let $A$ be a subnormal operator, and assume that $A^2$ is quasinormal. Does it follow that $A$ is quasinormal? In this paper, we give an affirmative answer to this question. In fact, we prove more general result that subnormal $n$th roots of quasinormal operators are quasinormal.

Theorems & Definitions (13)

• Theorem 1.2
• Theorem 1.3: Embry's characterization Embry73
• Theorem 2.1
• Example 2.2
• Theorem 2.3: Löwner-Heinz inequality He51Lo34
• Theorem 2.4: Han80uch93
• proof : First proof of Theorem \ref{['maintw']}
• proof : Second proof of Theorem \ref{['maintw']}
• Theorem 4.1
• Theorem 4.2