Is the differential operator $D:P_n \to P_n$ is reducible? Find an element of $P_n$ that is of period $n+1$ under $D$. Here $D$ is Differential operator and $P_n$ is the vector space of all polynomials of degree $n$
Well I know that an operator $D$ is reducible if it has non-trivial $D$-invariant subspace. Further an operator has non-trivial $D$-invariant subspace if its characteristics polynomial splits. I know that for $D$ its characteristics polynomial is $x^{n+1}$.
But I don't think so $D$ has such a subspace.
Am i correct? How to find the element of period $n+1$.