In $$ L^{2} (\mathbb{R}^2, e^{{-x^2}-y^{2}} dx dy)$$ with subspace $D$ of finite linear combinations of $g_m=(x+iy)^m$ , $m\neq 0$ and integer $(g_0=1)$.
I need to show $\langle g_a|g_b \rangle=0$ if $a\neq b$
Having $$L=\frac{1}{2}\left(\frac{\partial}{\partial x}-\frac{\partial}{\partial y}\right), M=M_{x+iy} $$ Acting on $D$. And I need to show $\langle h_1|Mh_2 \rangle=\langle Lh_1|h_2 \rangle$ if $h_1,h_2 \in D$.
And lastly I need to show $LM-ML=I$ on $D$.
I have this problem , which has baffled me for some time now... I do not know how to tackle it and how to proceed. I would really appreciate help.
Here is my progress: I was able to sort of do first part, but I am not sure it is correct. So I took $x+iy= re^{i\theta}$ and integrated $(re^{i\theta})^m$ with conjugate of $(re^{i\theta})^n$ and I got $0$ if $a\neq b$. But for the next two part I have no idea how..
Thank you!