  It is well-known that, due to the interaction between the spin and the
magnetic field, the two-dimensional Pauli operator has an eigenvalue $0$ at the
threshold of its essential spectrum. We show that when perturbed by an
effectively positive perturbation, $V$, coupled with a small parameter
$\varepsilon$, these eigenvalues become resonances. Moreover, we derive
explicit expressions for the leading terms of their imaginary parts in the
limit $\varepsilon\searrow 0$. These show, in particular, that the dependence
of the imaginary part of the resonances on $\varepsilon$ is determined by the
flux of the magnetic field. The cases of non-degenerate and degenerate zero
eigenvalue are treated separately. We also discuss applications of our main
results to particles with anomalous magnetic moments.
