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.