  We consider a compact Riemann surface $R$ of arbitrary genus, with a finite
number of non-overlapping quasicircles, which separate $R$ into two subsets: a
connected Riemann surface $\Sigma$, and the union $\mathcal{O}$ of a finite
collection of simply-connected regions. We prove that the Schiffer integral
operator mapping the Bergman space of anti-holomorphic one-forms on
$\mathcal{O}$ to the Bergman space of holomorphic forms on $\Sigma$ is an
isomorphism. We then apply this to prove versions of the Plemelj-Sokhotski
isomorphism and jump decomposition for such a configuration. Finally we obtain
some approximation theorems for the Bergman space of one-forms and Dirichlet
space of holomorphic functions on $\Sigma$ by elements of Bergman space and
Dirichlet space on fixed regions in $R$ containing $\Sigma$.
