# Toeplitz Order

## TL;DR

This paper develops a systematic function theoretic framework for Toeplitz operators in Harmonic Analysis, introducing a partial order on inner functions to advance understanding of classical Uncertainty Principle problems.

## Contribution

It introduces a new partial order on inner functions induced by Toeplitz operators, linking operator theory with classical harmonic analysis problems.

## Key findings

- Established connections between the new order and classical problems
- Provided insights into the structure of inner functions via Toeplitz operators
- Outlined future research directions in the area

## Abstract

A new approach to problems of the Uncertainty Principle in Harmonic Analysis, based on the use of Toeplitz operators, has brought progress to some of the classical problems in the area. The goal of this paper is to develop and systematize the function theoretic component of the Toeplitz approach by introducing a partial order on the set of inner functions induced by the action of Toeplitz operators. We study connections of the new order with some of the classical problems and known results. We discuss remaining problems and possible directions for further research.