Bergman projections, Toeplitz operators, weighted norms and periodic domains.

I. In the first part we consider spaces with upper doubling radial weights $\omega\in \widehat{\mathcal{D}}$ on the unit disc $\mathbb{D}$ of the complex plane. Given a weight $\omega$, let $P_\omega$ be the Bergman projection in the space $L_\omega^2$, endowed with the $\omega$-weighted area measure. Under the general assumption that either $\omega$ or $\nu$ belongs to $\widehat{\mathcal{D}}$, we give several characterizations of pairs $( \omega, \nu)$ such that $P_\omega$ is a bounded operator from $L^\infty_{\widehat{\nu}}$ onto its closed subspace $H^\infty_{\widehat{\nu}}$ consisting of analytic functions. We also solve the analogous problems for the boundedness of $P_\omega$ from $L^\infty_{\widehat{\nu}}$ onto the corresponding weighted Bloch type spaces and study similar questions for exponentially decreasing radial weights. This is a report on a joint work with \'Alvaro Miguel Moreno and Jos\'e \'Angel Pel\'aez, Universidad de M\'alaga.

II. In the second part 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 $T_{ \sf a}$.

报告人简介：

Taskinen, Senior University Lecturer at the Department of Mathematics and Statistics, University of Helsinki, the recipient of Ramsay Prize, Nevanlinna Jubileum Prize and Väisälä Prize.