# Between the von Neumann inequality and the Crouzeix conjecture

## TL;DR

This paper introduces a family of deformed numerical ranges $W^\rho(T)$ for operators, exploring their properties, including convexity, spectral containment, and dilation relations, extending classical numerical range concepts.

## Contribution

It defines and analyzes the new deformed numerical range $W^\rho(T)$, connecting it to spectral properties and dilation theory, and generalizing existing concepts like the numerical range.

## Key findings

- $W^\rho(T)$ is convex and contains the spectrum of $T$.
- $W^\rho(T)$ is decreasing in $\rho$ and coincides with the numerical range at $\rho=2$.
- $W^\rho(T)$ is contained in the unit disc iff $T$ has a $\rho$-unitary dilation.

## Abstract

A new concept of a deformed numerical range $W^\rho(T)$ is introduced. Here $T$ is a bounded linear operator or a matrix and $ \rho \in[1,+\infty)$ is a parameter. Each $W^\rho(T)$ is a closed convex set that contains the spectrum of $T$. Furthermore, $W^\rho(T)$ is decreasing with respect to $ \rho $ and $W^2(T)$ coincides with the numerical range. It is also shown that $W^\rho(T)$ is contained in the closed unit disc if and only if $T$ has a $\rho$ unitary dilation in the sense of N\'agy-Foia\c s. The spectral constants of $W^\rho(T)$ are investigated, it is shown that it is monotone and continuous with respect to the parameter $ \rho $.