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.
