# Asymptotic expantion of covariant symbol on the complex unit sphere

## TL;DR

This paper derives an asymptotic expansion for the Berezin transform on the complex unit sphere using a complete family of functions, and proves an Egorov-type theorem for covariant symbols of pseudo-differential operators.

## Contribution

It introduces a new asymptotic expansion for the Berezin transform on the sphere and establishes an Egorov-type theorem for covariant symbols in this context.

## Key findings

- Asymptotic expansion for Berezin transform derived
- Egorov-type theorem proved for covariant symbols
- Analysis involves asymptotic behavior of functions in a complete family

## Abstract

Starting from a complete family (not defined by the reproducing kernel) for the unit sphere $\mathbf S^n$ in the complex $n$-space $\mathbb C^n$, we obtain an asymptotic expansion for the associated Berezin transform. The proof involves the computation of the asymptotic behaviour of functions in the complete family. Furthermore, we prove an Egorov-type theorem for the covariant symbol related to a pseudo-differential operator on $L^2(\mathbf S^n)$.