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I have a question related to the Jordan-Chevalley Decomposition but I am also wondering about the general case.

I have that V is a finite dimensional vector space over $\mathbb{C}$ and $T:V\rightarrow V $linear. $T = D_{T}+N_{T}$ with $D_{T}$ diagonalizable and $N_{T}$ nilpotent and $D_{T}N_{T}=N_{T}D_{T}$.

I have already proved that if $W\subset V$ is T-Invariant then W is also $D_{T}$-Invariant and $N_{T}$-Invariant.

However, what I am struggling to prove is that if we were to restrict the domains of these functions then $T\mid_{w} = D_{T}\mid_{w}+N_{T}\mid_{w}$

Finally, in general, for arbitrary $T:V\rightarrow V $linear with $T = A + B$, A&B also linear, when is it that $T\mid_{w} = A\mid_{w}+B\mid_{w}$ for $W\subset V$

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You struggle witj a triviality.

If $w\in W$ then $$ T|_W(w)=T(w)=A(w)+B(w)=A|_W(w)+B|_W(w)$$

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