Let R \mathfrak {R} be a compact Riemann surface of finite genus g > 0 \mathfrak {g}>0 and let Σ \Sigma be the subsurface obtained by removing n ≥ 1 n\geq 1 simply connected regions Ω 1 + , ... , Ω n + \Omega _1^+, \dots , \Omega _n^+ from R \mathfrak {R} with non-overlapping closures. Fix a biholomorphism f k f_k from the unit disc onto Ω k + \Omega _k^+ for each k k and let f = ( f 1 , ... , f n ) \mathbf {f}=(f_1, \dots , f_n) . We assign a Faber and a Grunsky operator to R \mathfrak {R} and

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... f \mathbf {f} when all the boundary curves of Σ \Sigma are quasicircles in R \mathfrak {R} . We show that the Faber operator is a bounded isomorphism and the norm of the Grunsky operator is strictly less than one for this choice of boundary curves. A characterization of the pull-back of the holomorphic Dirichlet space of Σ \Sigma in terms of the graph of the Grunsky operator is provided.