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In the present paper, we prove an abstract functional analytic criterion fora class of linear partial differential operators acting on a domain$\Omega\subseteq\Bbb R^n$ which are elliptic in the interior to have compactresolvent. This extends known results for magnetic Schr\"{o}dinger operators tomore general differential operators. We point out the relationship between theDirac operator in real dimension two and the $\bar\partial$-Laplacian on acertain weighted space on $\Bbb C$ and we use this connection to prove anon-compactness result for its resolvent.

On the resolvent of the Dirac operator in $\Bbb R^2$