Compact differences of composition operators

TL;DR

This paper investigates the compactness of differences of composition operators induced by linear-fractional maps on the unit ball, revealing that non-trivial compact differences do not occur on Hardy or weighted Bergman spaces.

Contribution

It establishes that the difference of two such composition operators cannot be non-trivially compact, highlighting geometric aspects of the inducing maps.

Findings

Difference of composition operators cannot be non-trivially compact

Results apply to Hardy and weighted Bergman spaces

Emphasizes geometric properties of inducing maps

Abstract

When $\varphi$ and $\psi$ are linear-fractional self-maps of the unit ball $B_N$ in ${\mathbb C}^N$, $N\geq 1$, we show that the difference $C_{\varphi}-C_{\psi}$ cannot be non-trivially compact on either the Hardy space $H^2(B_N)$ or any weighted Bergman space $A^2_{\alpha}(B_N)$. Our arguments emphasize geometrical properties of the inducing maps $\varphi$ and $\psi$.