Closed linear operator

Question

I am having some trouble in showing the following map is closed:

For $f\in L^2(\mathbb{R^2})$ with $(x+iy)f(x,y)\in L^2(\mathbb{R^2})$, $M(f)(x,y)=(x+iy)f(x,y)$.

I am also asked to find the resolvent set.

So far I have found a subsequence in $L^2$, from the definition of closed linear operator and convergence in $L^p$ such that $f_{n_k}(w)$ -> $f(w)$ almost everywhere. From here I get a big stuck. Could someone please help me out? Thanks. Lachlan

Answer 1

Hint: if $I_r$ is the indicator function of the ball of radius $n$, $I_r f \to f$ in $L^2$ and $I_r f \in D(M)$. If $f_n \to f$ and $M f_n \to g$, what can you say about $M(I_r f)$?