On the spectral character of Toeplitz operators on planar regions

Self-adjoint Toeplitz operators on multiply connected planar regions are investigated using theta functions on the double. An explicit resolvent form for self-adjoint Toeplitz operators on a Hardy space associated with any representing measure on a $g$-holed planar region is given via reproducing kernels in terms of theta functions on $\mathbb {C}^g$. This resolvent formula is a generalization of an analogous formula obtained by K. F. Clancey (1991) for the case of harmonic measure. Applications of this resolvent form to the spectral type of the self-adjoint Toeplitz operators are described.