  We consider discretizations of the hyper-singular integral operator on closed
surfaces and show that the inverses of the corresponding system matrices can be
approximated by blockwise low-rank matrices at an exponential rate in the block
rank. We cover in particular the data-sparse format of $\mathcal{H}$-matrices.
We show the approximability result for two types of discretizations. The first
one is a saddle point formulation, which incorporates the constraint of
vanishing mean of the solution. The second discretization is based on a
stabilized hyper-singular operator, which leads to symmetric positive definite
matrices. In this latter setting, we also show that the hierarchical Cholesky
factorization can be approximated at an exponential rate in the block rank.
