On $n$th roots of bounded and unbounded quasinormal operators

TL;DR

The paper addresses whether subnormal (or related) operators whose $n$th powers are quasinormal must themselves be quasinormal. It develops two main directions: (i) for class $A$ bounded operators, if $T^n$ is quasinormal then $T$ is quasinormal, and (ii) for closed densely defined subnormal (unbounded) operators, if $T^n$ is quasinormal then $T$ is quasinormal. The authors advance the toolkit with an intertwining theorem and a moment-problem framework to handle unbounded cases, yielding a robust, general proof strategy. They also show that non-normal quasinormal operators with a quasinormal $n$th root can possess many non-quasinormal $n$th roots, highlighting intrinsic limitations of root-closure phenomena. Together, these results significantly extend previous work on spectral and monotonic properties across bounded and unbounded operator classes and illuminate the landscape of possible $n$th roots in this setting.

Abstract

In a recent paper [9], R. E. Curto, S. H. Lee and J. Yoon asked the following question: Let $T$ be a subnormal operator, and assume that $T^2$ is quasinormal. Does it follow that $T$ is quasinormal?. In [36] we answered this question in the affirmative. In the present paper, we will extend this result in two directions. Namely, we prove that both class A $n$th roots of bounded quasinormal operators and subnormal $n$th roots of unbounded quasinormal operators are quasinormal. We also show that a non-normal quasinormal operator having a quasinormal $n$th root has a non-quasinormal $n$th root.

Figures (1)

• Figure 1: Inclusion relations between the classes of operators under consideration.

Theorems & Definitions (30)

• Theorem 1.1: P-S21
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• proof
• Theorem 1.6
• proof
• Theorem 1.7
• Theorem 2.1: Embry73jabl14
• Theorem 2.2: Ito02; cf. Ito99 and Yama99