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

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.