# Which multiplication operators are surjective isometries?

## TL;DR

This paper characterizes when multiplication operators on certain Banach spaces of continuous functions are surjective isometries, identifying conditions under which they are only scalar multiples of the identity, using geometric and operator-theoretic tools.

## Contribution

It provides new sufficient conditions involving the inclusion operator and geometric properties of the space to identify surjective isometric multiplication operators.

## Key findings

- Surjective isometries are scalar multiples of the identity under certain conditions.
- Conditions involve the properties of the inclusion operator into continuous functions.
- Birkhoff Orthogonality is a key tool in the analysis.

## Abstract

Let $\mathbf{F}$ be a Banach space of continuous functions over a connected locally compact space $X$. We present several sufficient conditions on $\mathbf{F}$ guaranteeing that the only multiplication operators on $\mathbf{F}$ that are surjective isometries are scalar multiples of the identity. The conditions are given via the properties of the inclusion operator from $\mathbf{F}$ into $\mathcal{C}\left(X\right)$, as well as in terms of geometry of $\mathbf{F}$. An important tool in our investigation is the notion of Birkhoff Orthogonality.