Elliptic functions, Floquet transform and Bergman spaces on doubly periodic domains

Abstract

We study Bergman spaces A^2(D), their kernels and Toeplitz operators on unbounded, doubly periodic domains D in the complex plane. We establish the mapping properties of the Floquet transform operator defined in A^2(D) and derive a general formula connecting the Bergman kernel and projection of the domain D to a kernel and projection on the bounded periodic cell B. As an application, we prove, for Toeplitz operators T_a with doubly periodic symbols, a spectral band formula, which describes the spectrum and essential spectrum of T_a in terms of the spectra of a family of Toeplitz-type operators on the cell B. Technical challenges arise from the fact that double quasiperiodic boundary conditions have to be taken into account in the definitions of the spaces and operators on the periodic cell B. This requires novel operator theoretic tools, which are based on modifications of certain elliptic functions, e.g. the Weierstrass p-function.