On Bergman-Toeplitz operators in periodic planar domains

TL;DR

This paper analyzes the spectra of Toeplitz operators with periodic symbols on unbounded periodic domains, introducing Floquet techniques and constructing operators with prescribed spectral properties, including disjoint essential spectrum components.

Contribution

It develops Floquet-transform methods for Bergman Toeplitz operators in periodic domains and demonstrates how to construct operators with specific spectral features.

Findings

Established a band-gap-spectrum formula for Toeplitz operators in periodic domains.

Constructed examples of Toeplitz operators with essential spectrum containing disjoint components.

Provided a systematic approach to realize spectral properties using Toeplitz operators on the unit disc.

Abstract

We study spectra of Toeplitz operators $T_a $ with periodic symbols in Bergman spaces $A^2(\Pi)$ on unbounded periodic planar domains $\Pi$, which are defined as the union of infinitely many copies of the translated, bounded periodic cell $\varpi$. 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 $T_a$ in terms of the spectra of a family of Toepliz-type operators $T_{a,\eta}$ in the cell $\varpi$, where $\eta$ is the so-called Floquet variable. As an application, we consider periodic domains $\Pi_h$ containing thin geometric structures and show how to construct a Toeplitz operator $T_{\sf a}: A^2(\Pi_h) \to A^2(\Pi_h)$ such that the essential spectrum of $T_{\sf a}$ contains disjoint components which approximatively coincide with any given finite set of real numbers. Moreover, our method provides a systematic and illustrative way how to construct such examples by using Toeplitz operators on the unit disc $\mathbb{D}$ e.g. with radial symbols. Using a Riemann mapping one can then find a Toeplitz operator $T_a : A^2(\mathbb{D}) \to A^2(\mathbb{D})$ with a bounded symbol and with the same spectral properties as $T_{\sf a}$.