I've been working through some problems set by my University over the past few years, and have encountered this problem.
Problem Let $n > 1$ and let $V_n$ be the subspace of $\mathbb{R}[x, y]$ of dimension $n+1$ consisting of homogeneous polynomials of degree $n$, that is, the subspace spanned by $x^n, x^{n-1}y, \ldots y^n$. Let $P$ and $Q$ be the linear transformations on $V_n$ defined for $f$ in $V_n$ by
$ Pf=x\frac{\delta f}{\delta y} \quad \quad \quad Qf=y\frac{\delta f}{\delta x} $
What are the minimum polynomials of $P$, $Q$ and ($PQ-QP$) and which of these are diagonalisable?
Progress
We denote the minimum polynomial of any endomorphism $T$ on $V_n$ to be $m_T(X)$.
Let us first consider $P$; we look to find $m_P$ such that $m_P(P)=0$. Certainly the the polynomial $m_P(X)=X^{n+1}$ satisfies the condition as this reduces all the $y$ terms to zero. As such, $m_P(X)|X^{n+1}$. Similarly $m_Q(X)|X^{n+1}$.
Can anyone offer assistance with progressing further? I realise I haven't made much ground. Regards.