On a relationship between orthogonal projections and Toeplitz operators on poly-Bergman spaces of the upper half-plane: vertical symbols

Abstract

In the context of studying $C^{\ast }$-algebras generated by Toeplitz operators acting on the poly-Bergman space ${\mathcal{A}}_n^2(\hat{I})$ of the upper half-plane $\hat{I}$, we introduce a system of all-but-one orthogonal projections in generic position. We show that the $C^{\ast }$-algebra generated by these orthoprojections is closely related to the $C^{\ast }$-algebra generated by all Toeplitz operators with vertical symbols satisfying boundary conditions. This result suggests a new approach in the study of Toeplitz operators acting on other reproducing kernel Hilbert spaces. Furthermore, the range of one of the orthoprojections herein has a reproducing kernel expressed in terms of the digamma and the Nielsen's beta functions. The harmonic function also emerges in this development.