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

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 $\bar{\mathbb{D}}$ is a spectral set for $T$; $T$ can be dilated to a Hilbert space isometry; $\bar{\mathbb{D}}$ 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 ${\bar{D}}_R$ 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 $\sup \{r>0:{\bar{D}}_R\text{ is a spectral set for }{\mathcal{F}}_r\}=\frac{R}{3}$. We found several new characterizations for a Hilbert space in terms of spectral set and complete spectral set for different operators.