Indefinite Hardy Spaces on Finite Bordered Riemann Surfaces

It is well known that one may study Hardy spaces with the domain a finite bordered Riemann surface rather than the unit disk. However, for domains with more than one boundary component it is natural to consider, besides the usual positive definite inner product on H₂, indefinite inner products obtained by picking up different signs (or in the vector-valued case, different signature matrices) when integrating over different components. In this paper we obtain a necessary and sufficient condition for such an indefinite inner product to be nondegenerate and show that when this condition is satisfied we actually get a Krein space. Furthermore, each holomorphic mapping of the finite bordered Riemann surface onto the unit disk (which maps boundary to boundary) determines an explicit isometric isomorphism between this space and a usual vector-valued Hardy space on the unit disk with an indefinite inner product defined by an appropriate hermitian matrix. As is usual when studying Hardy spaces on a multiply connected domain, the elements of the space are sections of a vector bundle rather than functions. The main point is to construct an appropriate extension of this bundle to the double of the finite bordered Riemann surface and to use Cauchy kernels for certain vector bundles on a compact Riemann surface.