Invariant subspaces of derivative transformation and integral transformation on the linear space of polynomial ring.

Question

$\mathbb{F}$ is a field,$V=\mathbb{F}[x]$,$D$ and $S$ are derivative transformation and integral transformation on$V$.

$$D:p(x)\mapsto p'(x).$$ $$S:p(x)\mapsto \int_{0}^{x}p(t)\mathrm{d}t.$$

I want to know something about their invariant subspaces.

It's easy to find there are no such invariant subspaces of $S$ with finite dimension.But what about infinite dimensions?Do they exist and how to find them?Any help will be thanked.

Answer 1

The nontrivial invariant subspaces under $D$ are exactly the subspaces of polynomials of degree $\le n$. The nontrivial invariant subspaces under $S$ are exactly the subspaces of polynomials divisible by $x^n$ for each $n \in \mathbb{Z}_{\ge 0}$. Put together, there are no nontrivial invariant subspaces for the combined actions of $D$ and $S$.