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Let D be a bounded, convex domain in Ã¢Â„Â‚n, and suppose that ÃÂ†:DÃ¢Â†Â’D is holomorphic. Assume that ÃÂˆ:DÃ¢Â†Â’Ã¢Â„Â‚ is analytic, bounded away from zero toward the boundary of D, and not identically zero on the fixed point set of D. Suppose also that the weighted composition operator WÃÂˆ,ÃÂ† given by WÃÂˆ,ÃÂ†(f)=ÃÂˆ(fÃ¢ÂˆÂ˜ÃÂ†) is compact on a holomorphic, functional Hilbert space (containing the polynomial functions densely) on D  with reproducing kernel K satisfying K(z,z)Ã¢Â†Â’Ã¢ÂˆÂž as zÃ¢Â†Â’Ã¢ÂˆÂ‚D. 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 ÃÂ† has a unique fixed point in D. We apply this result by making a reasonable conjecture about the spectrum of WÃÂˆ,ÃÂ† based on previous results

Compact Weighted Composition Operators and Fixed Points in Convex Domains