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arXiv · 1907.06228

Between the von Neumann inequality and the Crouzeix conjecture

Patryk Pagacz·Paweł Pietrzycki·Michał Wojtylak
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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$.

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Patryk Pagacz, Paweł Pietrzycki, Michał Wojtylak. 2020-05-21. Between the von Neumann inequality and the Crouzeix conjecture. https://doi.org/10.1016/j.laa.2020.07.002

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