let $V$ be a vector space of final dimension above $F$ and $T,S\colon V \to V$ diagonalizable linear operators for which holds $ST=TS$. I need to prove $S+T$, $TS$ are also diagonalizable.
For that I have the following guidance:
I need to show that from $ST = TS$ can be infered that $T, S$ simultaneously diagonalizable, i.e., there is a base $B$ of $V$ in which both operators represented by diagonal matrices simultaneously. That I need to show in the following way:
firstly, I need to show that for each $\lambda$ holds $S(T-\lambda I)=(T-\lambda I)S$
secondly, I need to show that $W_\lambda = ker(T-\lambda I)$ is invariant subspace.
finally, show by induction on the space dimension $n$ that $T, S$ simultaneously diagonalizable, i.e show that $\dim(W_\lambda)\le\dim(V)$ and to use the induction assumption to get that the reductions of $T,S$ on $W_\lambda$, i.e $T_{|W_{\lambda }}$, $T_{|W_{\lambda }}$ simultaneously diagonalizable.
In the induction step I need to unite the bases of the sub-spaces of $W_\lambda$ which I got previously.
I easily did the first two points:
And I've got stuck in further attempts to continue.
Somebody helped me out with it, but I didn't understand his solution, because he wrote statement 4, which he never mentioned before:
Can anyone help me figure it out or maybe correct the proof according to the written steps ?