Let $X$ be a Banach space and $T: X \rightarrow X$ be a bounded linear operator. Let $Y_1, Y_2$ be closed $T$-invariant subspaces of $X$. Let $T_i= T \big|_ {Y_i}$ for $i=1,2$. If $\sigma (T)= \sigma(T_1) \cup \sigma(T_2)$, where $\sigma (T)$ is the spectrum of $T$.
If $\lambda \in \mbox{iso} \sigma (T)$ then prove that $H_0 (\lambda I-T)= H_0(\lambda I-T_1) \oplus H_0(\lambda I-T_2)$,
where $H_0 (T) $ is the quasi-nilpotent part of $T$ which is defined as $H_0(T):= \{x \in X : \lim\limits_{n \rightarrow \infty} ||T^n x|| ^{\frac{1}{n}} =0 \}.$