  For $n$-normal operators $A$ [2, 4, 5], equivalently $n$-th roots $A$ of
normal Hilbert space operators, both $A$ and $A^*$ satisfy the
Bishop--Eschmeier--Putinar property $(\beta)_{\epsilon}$, $A$ 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.

Functional Analysis