Perturbations and Weyl's theorem

A Banach space operator $T$ is completely hereditarily normaloid, $T\in\mathcal{CHN}$, if either every part, and (also) $T_p^{-1}$ for every invertible part $T_p$, of $T$ is normaloid or if for every complex number $\lambda$ every part of $T-\lambda I$ is normaloid. Sufficient conditions for the perturbation $T+A$ of $T\in\mathcal{CHN}$ by an algebraic operator $A$ to satisfy Weyl's theorem are proved. Our sufficient conditions lead us to the conclusion that the conjugate operator $(T+A)^*$ satisfies $a$-Weyl's theorem.