Faber and Grunsky operators corresponding to bordered Riemann surfaces

Abstract:

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