Finite number of T-invariant subspaces iff the restriction has a finite number of T-invariant subspaces

linear algebra

Let $V$ a finite vector space over a field $F$ and let $T\in \mathcal L({V})$ an operator.

$p_m=p_1^{r_1}\cdots p_n^{r_n}$ the minimal polynomial of $T$ and $W_i=\ker(p_i^{r_i}(T))$

$V=\displaystyle \bigoplus_{i=1}^nW_i $ $\hspace{0.4cm}$ by the primary decomposition theorem.

Show that $T$ has a finite number of $T$-invariant subspaces iff $T|_{W_j}$
has a finite number of T-invariant subspaces for every $1 \leq j \leq n$

I was trying to prove it without using cyclic vector.

Best Answer

  • Check that the $W_i$ are $T$-invariant subspaces.
  • Check that if $S$ is a $T$-invariant subspace, then so is $S \cap W_i$.

Suppose there are finitely many $T$-invariant subspaces $S_1, \ldots, S_m$ of $V$. Then $S_1 \cap W_i, \ldots, S_m \cap W_i$ are the $T|_{W_j}$-invariant subspaces of $W_j$ (possibly with repeats).

If $S$ is a $T$-invariant subspace of $V$, then check that $$S = \bigoplus_{i=1}^n (S \cap W_i).$$ If each $W_i$ has finitely many $T|_{W_j}$-invariant subspaces, what does that say about the number of $T$-invariant subspaces of $V$?

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