On $n$th roots of normal operators

For $n$-normal operators $A$ [2, 4, 5], equivalently $n$-th roots $A$ of normal Hilbert space operators, both $A$ and $A^{\ast }$ satisfy the Bishop–Eschmeier–Putinar property $(\beta {)}_{ϵ}$, $A$ is decomposable and the quasi-nilpotent part $H_0(A-\lambda )$ of $A$ satisfies $H_0(A-\lambda {)}^{-1}(0)=(A-\lambda {)}^{-1}(0)$ for every non-zero complex $\lambda$. $A$ satisfies every Weyl and Browder type theorem, and a sufficient condition for $A$ to be normal is that either $A$ is dominant or $A$ is a class $\mathcal{A}(1,1)$ operator.