Direct sum of quasi-nilpotent part of an operator

Question

Let $X$ a Banach space and $T: X \rightarrow X$ be a bounded linear operator. Let $X=Y_1\oplus Y_2$ where $Y_1$ and $Y_2$ are closed $T$-invariant subspaces of $X$. Let $T_1 = T \big| _{Y_1}$ and $T_2 = T \big| _{Y_2}$. Then prove that $H_0(T)=H_0(T_1) \oplus H_0(T_2)$, where $H_0(T)$ is the quasi-nilpotent part of operator $T$ which is defined as $H_0(T)= \{ x \in X: \lim\limits_{n \rightarrow \infty} ||T^n x||^{\frac{1}{n}}=0\}.$