Commuting and diagonalizable endomorphisms

Question

If $f, g \in End(V)$ are diagonalizable and commuting, then how to prove that also $f \ \omicron \ g$ and $ f \pm g$ are diagonalizable?

I tried to use that $f, g$ are simultaneously diagonalizbale, but i didn't find a solution.

Answer 1

If $f$ and $g$ are simultaneously diagonalizable, then $\exists \varphi \in \text{GL}(V)$ such that $\varphi\circ f\circ\varphi^{-1} = a$ and $\varphi\circ g\circ\varphi^{-1} = b$ are diagonal.

Hence $f\circ g = \varphi^{-1}\circ((\varphi\circ f\circ\varphi^{-1})\circ (\varphi\circ g\circ\varphi^{-1}))\circ\varphi = \varphi^{-1}\circ( a\circ b)\circ\varphi$ is similar to the product of two diagonal matrices, hence it's diagonalizable.

Moreover, $f + g = \varphi^{-1}\circ(\varphi\circ f\circ\varphi^{-1})\circ\varphi + \varphi^{-1}\circ(\varphi\circ g\circ\varphi^{-1})\circ\varphi = \varphi^{-1}\circ a \circ\varphi + \varphi^{-1}\circ b\circ\varphi$

thus $f + g = \varphi^{-1}\circ(a+b)\circ\varphi$ is diagonalizable.