Between the von Neumann inequality and the Crouzeix conjecture

Patryk Pagacz, Paweł Pietrzycki, Michał Wojtylak

Published 2019-07-14Version 1

A new concept of a deformed numerical range $W_q(T)$, where $T$ is a bounded linear operator or a matrix and $q\in[0,2)$ is a parameter, is introduced. Each $W_q(T)$ is a closed convex set that contains the spectrum of $T$. Furthermore, $W_q(T)$ is decreasing with respect to $q$ and $W_1(T)$ is the numerical range. It is also shown that $W_q(T)$ is contained in the closed unit disc if and only if $T$ has a $2/(2-q)$ unitary dilation in the sense of N\'agy-Foias. Spectral constants of $W_q(T)$ are investigated.