Compact differences of composition operators

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

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