I haven't seen this question yet, so I want to ask this:
Let $W_1$ and $W_2$ be $T$-invariant subspaces, with minimal polynomials restricted to the subspaces $W_1$ and $W_2$ being $\mu(x)$ and $\gamma(x)$ respectively. Let the minimal polynomial of $T$ be $m(x)$.
I know that if $W_1+W_2=V$, $m(x)=\text{lcm}(\mu(x),\gamma(x))$. However, is there any condition that allows me to multiply the minimal polynomials? That is, are there any conditions on the subspaces $W_1$ and $W_2$ that allow me to say $m(x)=\mu(x) \gamma(x)$. I don't think that $W_1 \oplus W_2=V$ is strong enough, as mentioned in this post, because I need $\gcd(\mu(x),\gamma(x))=1$, but if $T$ is the identity operator, I can always take a basis for $V=\{b_1, b_2, \cdots b_n\}$, and let the cyclic subspace generated by $b_i$ be $Z[I, b_i]$. Clearly $V=Z[I, b_1] \oplus Z[I, b_2] \cdots \oplus Z[I, b_n]$, but the minimal polynomial of $I$ restricted to any $Z[I, b_i]$ is $(x-1)$, so their $\gcd$'s are $(x-1)$.
Is there any condition that allows me to say with certainity that $m(x)=\mu(x) \gamma(x)$? I was hoping for a stronger condition than simply saying that this will occur when either $\mu(x)$ or $\gamma(x)$ is irreducible.
