Between the von Neumann inequality and the Crouzeix conjecture
Patryk Pagacz, Paweł Pietrzycki, Michał Wojtylak
TL;DR
This work introduces the deformed numerical range $W^\rho(T)$, an intermediate spectral set interpolating between the von Neumann inequality and the Crouzeix conjecture by varying the dilation parameter $\rho$. It establishes core properties including spectrum containment, monotonicity and continuity in $\rho$, and a corresponding $\rho$-dependent spectral constant $\Psi_\rho(T)$ governing polynomial functional calculus. A key result is the equivalence between $\rho$-dilations and the unit disk containment $W^\rho(T)\subseteq\overline{\mathbb{D}}$, linking dilation theory with numerical-range based constants. The paper further relates $W^\rho(T)$ to established concepts such as the $B_\rho\otimes T$ dilation, the $q$-numerical range, the normalised numerical range, and the Davies–Wielandt shell, providing a cohesive framework to understand spectral constants as a continuous bridge between classical bounds.
Abstract
A new concept of a deformed numerical range $W^ρ(T)$ is introduced. Here $T$ is a bounded linear operator or a matrix and $ ρ\in[1,+\infty)$ is a parameter. Each $W^ρ(T)$ is a closed convex set that contains the spectrum of $T$. Furthermore, $W^ρ(T)$ is decreasing with respect to $ ρ$ and $W^2(T)$ coincides with the numerical range. It is also shown that $W^ρ(T)$ is contained in the closed unit disc if and only if $T$ has a $ρ$ unitary dilation in the sense of Nágy-Foia\c s. The spectral constants of $W^ρ(T)$ are investigated, it is shown that it is monotone and continuous with respect to the parameter $ ρ$.