A remark on non-surjective composition operators


Assume that A, B are uniform algebras on compact Hausdorff spaces X and Y, respectively, and ∂A, ∂B are the Šilov boundaries of A, B. Let T : A −1 → B−1 be a map with T1 = 1. We show that, if there exist constants α, β ≥ 1 such that β−1‖f·g−1‖ ≤ ‖Tf·(Tg) −1∥ ≤ α∥f·g−1∥ for all f, g ∈ A−1, then there is a non-empty closed subset Y 0 of ∂B and a surjective continuous map τ : Y 0 → ∂A such that (Figure presented.) for all f ∈ A−1 and all y ∈ Y 0. Moreover we give an example which shows that the multiple α 2 β in the above inequality is the best possible.

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Quaestiones Mathematicae