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

TL;DR

This paper explores the relationship between orthogonal projections and Toeplitz operators on poly-Bergman spaces of the upper half-plane, revealing new algebraic connections and explicit kernel representations involving special functions.

Contribution

It introduces a system of orthogonal projections in generic position and links their generated $C^*$-algebra to that of Toeplitz operators with vertical boundary symbols, offering a novel analytical approach.

Findings

The $C^*$-algebra generated by these projections relates closely to Toeplitz operators with boundary conditions.

The range of one projection has a reproducing kernel involving digamma and Nielsen's beta functions.

Harmonic functions play a role in the structure of these operators.

Abstract

In the context of studying $C^*$-algebras generated by Toeplitz operators acting on the poly-Bergman space $\mathcal{A}^2_{n}(\Pi)$ of the upper half-plane $\Pi$, we introduce a system of all-but-one orthogonal projections in generic position. We show that the $C^*$-algebra generated by these orthoprojections is closely related to the $C^*$-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.