Logarithms and exponentials in Banach algebras

TL;DR

This paper proves that invertible elements in complex Banach algebras have logarithms if their spectrum doesn't separate the plane, offering elementary proofs and applications to matrices over complex and real fields.

Contribution

Provides elementary, holomorphic-calculus-free proofs of the logarithm existence for invertible elements in Banach algebras and applies these results to matrices over complex and real fields.

Findings

Invertible elements with non-separating spectrum admit a logarithm.

Every invertible complex matrix has a logarithm.

Real matrices with positive determinant are products of two exponentials.

Abstract

Let $A$ be a complex Banach algebra. If the spectrum of an invertible element $a\in A$ does not separate the plane, then $a$ admits a logarithm. We present two elementary proofs of this classical result which are independent of the holomorphic functional calculus. We also discuss the case of real Banach algebras. As applications, we obtain simple proofs that every invertible matrix over $\mathbb C$ has a logarithm and that every real matrix $M$ in $M_n(\mathbb R)$ with $\det M>0$ is a product of two real exponential matrices.