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Compact Weighted Composition Operators and Fixed Points in Convex Domains

Abstract

Let be a bounded, convex domain in , and suppose that is holomorphic. Assume that is analytic, bounded away from zero toward the boundary of , and not identically zero on the fixed point set of . Suppose also that the weighted composition operator given by is compact on a holomorphic, functional Hilbert space (containing the polynomial functions densely) on with reproducing kernel satisfying as . We extend the results of J. Caughran/H. Schwartz for unweighted composition operators on the Hardy space of the unit disk and B. MacCluer on the ball by showing that has a unique fixed point in . We apply this result by making a reasonable conjecture about the spectrum of based on previous results.

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Correspondence to Dana D Clahane.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Clahane, D.D. Compact Weighted Composition Operators and Fixed Points in Convex Domains. Fixed Point Theory Appl 2007, 028750 (2007). https://doi.org/10.1155/2007/28750

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