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 $T=T^{\prime }\oplus T^{\prime \prime }$, where $T^{\prime }$ is normal and $T^{\prime \prime }$ is nilpotent of nil-index at most $\min \{n,k+1\}$, with either summand potentially absent.