From Linear Algebra from Hoffman and Kunze's book, page 220.
Let T be a linear operator on the finite-dimensional vector space V over the field F. Let $m_T=p_1^{r_1}\dots p_k^{r_k}$ be the minimal polynomial for $T$, where $p_{i}$ are distinct irreducible monic polynomials over $\mathbb{F}$ and $r_{i}$ are positive numbers. Let $W_{i}$ be the null space of $p_{i}(T)^{r_{i}},\,\,i=1,\dots, k$. Then
(i) $V=W_1\oplus \cdots \oplus W_k$;
(ii) each $W_{i}$ is $T$-invariant;
(iii) if $T_{i}$ is the operator induced on $W_{i}$ by $T$, then the minimal polynomial for $T_{i}$ is $p_{i}^{r_{i}}$.
Is there a version of Primary Decomposition Theorem for infinite-dimensional vector spaces?