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

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.