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Let $V$ be a finite dimensional vector space over a field $K$ of characteristic zero, and let $T \in End(V)$. A subspace $W \subseteq V$ is $T$-invariant if $T(W) \subseteq W$; and $W$ is $T$-irreducible if it is T-invariant and the only T-invariant subspaces of $W$ are $0$ and $W$.

(a) Suppose that $T \in End(V)$ has order $n$ in $End(V)$ under composition, and that $W \subseteq V$ is a T-invariant subspace, Show that $W$ has a T-invariant complement $W'$.

(b). Under the hypotheses of $(a)$, show that $V$ can be written as the direct product of T-irreducible subspaces.

(c). What if $T$ does not have finite order in $End(V)$?

(d). What if $K$ does not have characteristic zero?

I was able to prove parts (a) and (b). However, I am not so sure about parts (c) and (d). Since my proof of parts (a) and (b) relied on the fact that $T$ had finite order and $K$ had characteristic zero, I assume that the results no longer hold. This all seems similar to to Maschke's theorem, but I am not familiar with representation theory.

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The matrix $T:=\pmatrix{1&1\\0&1}$ provides a counterexample for both c) and d).
Can you see how?

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  • $\begingroup$ Thank you for the response. It's not quite clear to me how this serves as a counterexample. Would you mind expanding a bit more? $\endgroup$
    – slowspider
    Commented Nov 28, 2021 at 2:06
  • $\begingroup$ Well, set $W:={\rm span}\pmatrix{1\\0}$. Show that it doesn't have a $T$-invariant complement. $\endgroup$
    – Berci
    Commented Nov 28, 2021 at 9:53

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