Let's first proof T is diagonalizable$\displaystyle \Rightarrow$ V = E1 + E2.
If T is diagonalizable, it can be written as $\displaystyle PDP^{-1}$, where D is a diagonal matrix with its eigenvalues on the main diagonal, and P represents a basis of eigenvectors.
That basis spans V.
Therefore V is the direct sum of the eigen spaces.
Now let's proof $\displaystyle V = E_1 + E_2$ $\displaystyle \Rightarrow$ T is diagonalizable.
This means that that basis of $\displaystyle E_1$ together with the basis of $\displaystyle E_2$ span V.
Put those bases next to each other in a matrix P, and put the eigenvalues diagonally in a corresponding matrix D, and you have that
$\displaystyle TP = PD$
$\displaystyle T = PDP^{-1}$
So T is diagonalizable.
Therefore T is diagonalizable iff $\displaystyle V = E_1+ E_2 \qquad \blacksquare$