# Note on Wermuth's theorem on commuting operator exponentials

## TL;DR

This paper explores conditions under which commuting operator exponentials imply the commutativity of the operators themselves, extending Wermuth's theorem to Banach and Hilbert spaces with new proofs.

## Contribution

It applies Wermuth's theorem to establish a criterion for operator commutativity via exponentials, providing an alternative proof for Hilbert spaces.

## Key findings

- If $A$ has a $2 extpi i$-congruence free spectrum, then $e^A B = B e^A$ iff $AB=BA$.
- The result extends to Banach spaces and offers a new proof for Hilbert spaces.
- Provides insights into the relationship between operator spectra and exponential commutativity.

## Abstract

We apply Wermuth's theorem on commuting operator exponentials to show that if $A, B \in B(X)$, $X$ being Banach space and $A$ of $2\pi i$-congruence free spectrum, then $e^A B = B e^A$ if and only if $AB=BA$. We employ this observation to provide alternative proof of similar result by Chaban and Mortad, applicable for $X$ being a Hilbert space.