# The mean transform and the mean limit of an operator

## TL;DR

This paper investigates the iterative behavior of the mean transform of bounded linear operators on Hilbert spaces, establishing new bounds and relationships for spectral properties, especially for unilateral weighted shifts.

## Contribution

It introduces the concept of the mean limit of an operator, provides new estimates for numerical range and radius, and characterizes the spectral radius for unilateral weighted shifts.

## Key findings

- Derived bounds for numerical range and radius of the mean transform.
- Established the relationship between spectral radius and mean limit for weighted shifts.
- Identified the conditions under which the mean limit exists and its properties.

## Abstract

Let $T$ be a bounded linear operator on a Hilbert space $\mathcal{H}$, and let $T \equiv V|T|$ be the polar decomposition of $T$. The mean transform of $T$ is defined by $\widehat{T}:=\frac{1}{2}(V|T|+|T|V)$. In this paper we study the iterates of the mean transform and we define the mean limit of an operator as the limit (in the operator norm) of those iterates. We obtain new estimates for the numerical range and numerical radius of the mean transform in terms of the original operator. For the special class of unilateral weighted shifts we describe the precise relationship between the spectral radius and the mean limit, and obtain some sharp estimates.