Between the von Neumann inequality and the Crouzeix conjecture

Abstract

A new concept of a deformed numerical range Wq(T ), where T is a bounded linear operator or a matrix and q ∈ [0, 2) is a parameter, is introduced. Each Wq(T ) is a closed convex set that contains the spectrum of T . Furthermore, Wq(T ) is decreasing with respect to q and W1(T ) is the numerical range. It is also shown that Wq(T ) is contained in the closed unit disc if and only if T has a 2/(2−q) unitary dilation in the sense of Nágy-Foiaş. Spectral constants of WqT are investigated.