On roots of normal operators and extensions of Ando's Theorem

TL;DR

This work addresses when normality of a power $T^n$ forces normality of $T$ for generalized operator classes beyond paranormal, including $k$-paranormal, absolute-$k$-paranormal, and $k$-quasi-paranormal operators. It proves that for $k$-paranormal or absolute-$k$-paranormal $T$, $T^n$ normal implies $T$ is normal, extending Ando's theorem. In the separable case, a $k$-quasi-paranormal operator with a normal power decomposes as $T=T'\oplus T''$, where $T'$ is normal and $T''$ is nilpotent of nil-index at most $\min\{n,k+1\}$, with a fiberwise direct-integral proof and connections to the square-root structure of normal operators. The results combine spectral methods with a detailed structural analysis, clarifying how normality of powers governs the overall operator behavior in these extended classes.

Abstract

In this paper, we extend Ando's theorem on paranormal operators, which states that if $ T \in \mathfrak{B}(\mathcal{H}) $ is a paranormal operator and there exists $ n \in \mathbb{N} $ such that $ T^n $ is normal, then $ T $ is normal. We generalize this result to the broader classes of $ k $-paranormal operators and absolute-$ k $-paranormal operators. Furthermore, in the case of a separable Hilbert space $\mathcal{H}$, we show that if $ T \in \mathfrak{B}(\mathcal{H}) $ is a $ k $-quasi-paranormal operator for some $ k \in \mathbb{N} $, and there exists $ n \in \mathbb{N} $ such that $ T^n $ is normal, then $ T $ decomposes as $ T = T' \oplus T'' $, where $ T' $ is normal and $ T'' $ is nilpotent of nil-index at most $ \min\{n,k+1\} $, with either summand potentially absent.

Figures (1)

• Figure 1: The $n$-th root problem illustration.

Theorems & Definitions (28)

• Definition 1.1
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• proof
• Lemma 2.1
• proof
• proof : A second proof of Theorem \ref{['thm:quasinormal_nth_roots']}
• Theorem 2.4
• Example 2.1