"Let $H_0=\begin{pmatrix}i&0\\0&-i\end{pmatrix}$, suppose $A\in SU(2)$ commutes with $H_0$, it must preserves each eigen spaces for $H_0$, eigen spaces for $H_0$ are just $\mathbb{C}e_1$ and $\mathbb{C}e_2$ where $e_1=(i,0)$, $e_2=(0,i)$ i.e standard basis element for $\mathbb{C}^2$",
could any one tell me what does it mean by preservation of eigen space?