Which multiplication operators are surjective isometries?

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.

18 pages

References in corpus (4)

• Self-adjoint, unitary, and normal weighted composition operators in several variables
• Multiplication operators on the Bloch space of bounded homogeneous domains
• A note on completeness of weighted normed spaces of analytic functions
• Continuity and Holomorphicity of Symbols of Weighted Composition Operators