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

TL;DR

The paper investigates how foundational Hilbert-space equivalences among contraction, spectral set, complete spectral set, and dilation fail in the Banach-space setting. It leverages Bohr's theorem to connect spectral-set phenomena with minimal spectral sets for operator families $\mathcal{F}_r$, establishing that the closed disk $\overline{D}_R$ is the minimal spectral set for $\mathcal{F}_r$ iff $r=R/3$, and that the Bohr radius of $D_R$ equals $R/3$. Through explicit Banach-space counterexamples, it shows that norm-control does not force spectral-set implications, and that dilation and complete spectral-set properties are not equivalent in general Banach spaces. The results yield new Hilbert-space characterizations and clarify the scope of spectral-set theory beyond the Hilbert setting, linking dilation theory to Bohr-type phenomena in Banach spaces. Overall, the work delineates the precise boundaries between contraction, spectral behavior, and dilation in Banach spaces, enriching the theory with concrete counterexamples and cross-connecting classical results.

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.

Theorems & Definitions (29)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 2.1: H. Bohr, HB
• Theorem 2.2: Theorem 2, VEK VIM
• Theorem 2.3: Generalized Bohr's Theorem
• proof
• Proposition 2.4