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I'm trying to find a reference for the following theorem in linear algebra.

Theorem. Let $k$ be a field, let $\mathfrak{P} \subseteq k[x]$ be a set of representatives of the irreducible polynomials in $k[x]$ (for example, if $k$ is algebraically closed, then we may take $\mathfrak{P} = \{x - a \mid a \in k\}$), let $V$ be a vector space over $k$, and let $H \in \mathrm{End}(V)$. For $p(x) \in \mathfrak{P}$, we set $V_{p(x)} = \{a \in V \mid \text{there is }k \in \mathbb{N}\text{ such that } p(H)^ka = 0\}$. We are not requiring any vector space to be finite dimensional. Then the family of subspaces $(V_{p(x)})_{p(x) \in \mathfrak{P}}$ is linearly independent and $H(V_{p(x)}) \subseteq V_{p(x)}$ for $p(x) \in \mathfrak{P}$. Therefore, the space $W = \sum_{p(x) \in \mathfrak{P}}V_{p(x)}$ satisfies $H(W) \subseteq W$, and we can safely assume that $V = W$. We now assume we have a subspace $U$ of $V$ such that $H(U) \subseteq U$. Then $U = \sum_{p(x) \in \mathfrak{P}}U \cap V_{p(x)}$.

In summary, the conclusions of this theorem are:

  1. The family of subspaces $(V_{p(x)})_{p(x) \in \mathfrak{P}}$ is linearly independent;
  2. $H(V_{p(x)}) \subseteq V_{p(x)}$ for $p(x) \in \mathfrak{P}$;
  3. $U = \sum_{p(x) \in \mathfrak{P}}U \cap V_{p(x)}$.
Zero
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  • Where is this from? – blargoner Jul 25 '23 at 17:28
  • @blargoner It's usually stated in a similar way in elementary linear algebra books, usually near the section about primary-decomposition theorem. But I need that theorem in its full form. – Zero Jul 25 '23 at 18:53
  • Are you certain it's true? The closest partial result I found at a quick glance was Bourbaki Algebra II Chapter VII Section 2 Theorem 1, but that's stated for torsion modules and I don't think $V$ will be a torsion $k[x]$-module in general. – blargoner Jul 25 '23 at 21:50
  • @blargoner I'm mostly certain it's true, or something very close to it. But you are encouraged to try to find a counterexample. – Zero Jul 26 '23 at 00:21
  • @blargoner Actually, the subspace $W$ is a torsion $k[x]$-module, right? I think Bourbaki Algebra II Chapter VII Section 2 Theorem 1 is conclusion 1. of my theorem, right? How about 2. and 3. (this is the hardest, I think)? – Zero Jul 26 '23 at 00:37
  • 2 is easy since $H$ commutes with any polynomial in $H$. 3 in finite dimensions follows from the projections being polynomials in $H$ (which actually characterizes direct sums of generalized eigenspaces) but not sure about infinite case. – blargoner Jul 26 '23 at 03:28
  • @blargoner Let $u \in W$ and write $u = u_{p(x)1} + \dots + u{p(x)n}$, where $p(x)_i \in \mathcal{P}$ are different and $u{p(x)i} \in V{p(x)i}$. Can we conclude 3. by considering the space $V' = \mathrm{span}{H^ku{p(x)_i} \mid k \in \mathbb{N}, i = 1, \dots, n}$ which is necessarily finite-dimensional? – Zero Jul 26 '23 at 03:37

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