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We consider a compact Riemann surface $R$ of arbitrary genus, with a finitenumber of non-overlapping quasicircles, which separate $R$ into two subsets: aconnected Riemann surface $\Sigma$, and the union $\mathcal{O}$ of a finitecollection of simply-connected regions. We prove that the Schiffer integraloperator mapping the Bergman space of anti-holomorphic one-forms on$\mathcal{O}$ to the Bergman space of holomorphic forms on $\Sigma$ is anisomorphism. We then apply this to prove versions of the Plemelj-Sokhotskiisomorphism and jump decomposition for such a configuration. Finally we obtainsome approximation theorems for the Bergman space of one-forms and Dirichletspace of holomorphic functions on $\Sigma$ by elements of Bergman space andDirichlet space on fixed regions in $R$ containing $\Sigma$.

Schiffer comparison operators and approximations on Riemann surfaces bordered by quasicircles