Hermitian powers: A Müntz theorem and extremal algebras
Volume 146 / 2001
Studia Mathematica 146 (2001), 83-97
MSC: Primary 46H05.
DOI: 10.4064/sm146-1-6
Abstract
Given ${\mathbb S}\subset {\mathbb N}$, let $\widehat {{\mathbb S}}$ be the set of all positive integers $m$ for which $h^m$ is hermitian whenever $h$ is an element of a complex unital Banach algebra $A$ with $h^n$ hermitian for each $n\in {\mathbb S}$. We attempt to characterize when (i) $\widehat {{\mathbb S}}={\mathbb N}$, or (ii) $\widehat {{\mathbb S}}={\mathbb S}$. A key tool is a Müntz-type theorem which gives remarkable conclusions when $1\in {\mathbb S}$ and $\sum \{ 1/n:n\in {\mathbb S}\} $ diverges. The set $\widehat {{\mathbb S}}$ is determined by a single extremal Banach algebra $\mathop {\rm Ea}\nolimits ({\mathbb S})$. We describe this extremal algebra for various ${\mathbb S}$.
