Spectral set, complete spectral set and dilation for Banach space operators

TL;DR

This paper investigates spectral sets and dilations for Banach space operators, showing that classical equivalences in Hilbert spaces do not hold generally, and characterizes spectral sets using Bohr's theorem.

Contribution

It demonstrates the failure of classical spectral set equivalences in Banach spaces and provides new characterizations of spectral sets related to Bohr's radius.

Findings

Spectral set equivalences for Hilbert spaces do not extend to Banach spaces.

The minimal spectral set for operators with norm ≤ r is the disk of radius R/3 when r=R/3.

The Bohr radius R/3 is characterized as the spectral set threshold for Banach space operators.

Abstract

Famous results due to von Neumann, Sz.-Nagy and Arveson assert that the following four statements are equivalent; a Hilbert space operator $T$ is a contraction; the closed unit disk $\overline{\mathbb D}$ is a spectral set for $T$; $T$ can be dilated to a Hilbert space isometry; $\overline{\mathbb D}$ is a complete spectral set for $T$. In this article, we show by counter examples that no two of them are equivalent for Banach space operators. If $\mathcal F_r$ is the family of all Banach space operators having norm less than or equal to $r$ and if $D_R$ denotes the open disk in the complex plane with centre at the origin and radius $R$, then we prove by an application of Bohr's theorem that $\overline{D}_R$ is the minimal spectral set for $\mathcal F_r$ if and only if $r=\frac{R}{3}$. Also, we prove the equivalence of the following two facts: the Bohr radius of $D_R$ is $\frac{R}{3}$ and $\sup \{ r>0\,:\, \overline{D}_R \text{ is a spectral set for } \mathcal F_r \}=\frac{R}{3}$. We found several new characterizations for a Hilbert space in terms of spectral set and complete spectral set for different operators.