On Bergman–Toeplitz operators in periodic planar domains

Abstract We study spectra of Toeplitz operators with periodic symbols in Bergman spaces on unbounded singly periodic planar domains , which are defined as the union of infinitely many copies of the translated, bounded periodic cell . We introduce Floquet‐transform techniques and prove a version of the band‐gap‐spectrum formula, which is well‐known in the framework of periodic elliptic spectral problems and which describes the essential spectrum of in terms of the spectra of a family of Toepliz‐type operators in the cell , where is the so‐called Floquet variable. As an application, we consider periodic domains containing thin geometric structures and show how to construct a Toeplitz operator such that the essential spectrum of contains disjoint components which approximatively coincide with any given finite set of real numbers. Moreover, our method provides a systematic and illustrative way on how to construct such examples by using Toeplitz operators on the unit disc , for example, with radial symbols. Using a Riemann mapping, one can then find a Toeplitz operator with a bounded symbol and with the same spectral properties as .