There is no T-invariant subspace $U$ such that $\mathbb{R}^{3} = W\oplus U$

Question

Could someone give me a suggestion to solve the following problem problem?

PROBLEM. Let $T : \mathbb{R}^{3} \longrightarrow \mathbb{R}^{3}$ and $\beta$ a basis of $\mathbb{R}^{3}$ such that $$ \left[ T \right]_{\beta} = \begin{pmatrix} 2 & 0 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}$$ If $W = \operatorname{ker}(T -2I)$, proof that there is no T-invariant subspace $U$ such that $\mathbb{R}^{3} = W\oplus U$.