Skew adjoint operators from Schroedinger representation of Heisenberg algebra

Consider the Heisenberg algebra $h_3$ generated by $\{X,Y,T\}$ with $[X,Y]=T$ and all other Lie bracket vanishing.

Let $\pi_*$ be the irreducible representation of $h_3$ derived from a Schroedinger representation $\pi$ on $\mathcal H$ of the corresponding group.

My questions are

1) Is $\pi_{*}(X)(\mathcal H)\cap\pi_{*}(Y)(\mathcal H)$ dense in $\mathcal H$?

2) Is $\pi_{*}(XY^2)(\mathcal H)+\pi_{*}(YX^2)(\mathcal H)$ dense in $\mathcal H$?