Between the von Neumann inequality and the Crouzeix conjecture

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 ∈ [ 1 , + ∞ ) 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 ) coincid...
Ausführliche Beschreibung

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 ∈ [ 1 , + ∞ ) 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-Foiaş. The spectral constant of W ρ ( T ) is investigated, it is shown that it is monotone and continuous with respect to the parameter ρ. Ausführliche Beschreibung