On functional calculus for Hermitian elements of Banach algebras: the norm and spectral radius

TL;DR

This paper explores the properties of Hermitian elements in Banach algebras, establishing a link between spectral radius and norm, and characterizing universal symbols through positive definite functions.

Contribution

It introduces the concept of Hermitian elements in Banach algebras and characterizes universal symbols using positive definite functions, extending spectral theory.

Findings

Hermitian elements satisfy |a|=||a|| in Banach algebras

Universal symbols are characterized by positive definite functions

In Hilbert space, Hermitian elements coincide with selfadjoint operators

Abstract

Let $ A$ be a complex unital Banach algebra. An element $a \in A$ is said to be Hermitian, if $ \| \exp (ita) \| =1$ for all $t\in R$. In the case of the algebra of bounded linear operators in a Hilbert space this Hermitian property agrees with the ordinary selfadjointness. If $a \in A$ is Hermitian, then $|a|=||a||$, where $|a|$ denotes the spectral radius of $a$. A function $F: R\to \C$ is called the universal symbol if $ \|| F(a)||= |F(a)|$\ for each $ A$ and all Hermitian $a\in A$. We characterize universal symbols in terms of positive definite functions.