Self-Adjoint Toeplitz Operators Associated with Representing Measures on Multiply Connected Planar Regions and Their Eigenvalues

A direct calculation of the vector of Riemann constants corresponding to the marked double of a multiply connected planar region is given. The existence of eigenvalues of self-adjoint Toeplitz operators acting on Hardy spaces associated with non-negative representing measures on 1-holed planar regions is established in the case where there exists one bounded component in the complement of the essential range of the symbol $phi $ of the operator. The analysis is done by using the zeros of translations of theta functions restricted to $mathbb R $ in $mathbb C $ .