Toeplitz Operators on Multiply Connected Domains and Theta Functions

The Fredholm spectral picture of Toeplitz operators acting on the least harmonic majorant Hardy space of a multiply connected planar domain as described by M.B. Abrahamse is refined. This is accomplished by viewing the planar domain as a domain on its double and applying the methods of Hilbert barrier problems associated with divisors as developed by R.N. Abdulaev, N. Koppelman, Yu.L. Rodin and E.I. Zverovich. In essence the results on barrier problems are obtained by reducing to the classical Riemann-Roch Theorem and the Riemann Singularity Theorem for theta functions. The barrier problems encountered are associated with the critical Green’s divisor and the results are considerably enhanced by the work on theta functions by J.D. Fay.