Compact weighted composition operators and fixed points in convex domains

Details Compact weighted composition operators and fixed points in convex domains Compact weighted composition operators and fixed points in convex domains

2005-12-02

arxiv.org/abs/math/0512044v3

Mathematics

Functional Analysis

10 pages. Corrected a few typographical errors and an error in one step of the main result's proof. This paper was presented i

Scientific paper

We extend a classical result of Caughran/Schwartz and another recent result of Gunatillake by showing that if D is a bounded, convex domain in n-dimensional complex space, m is a holomorphic function on D and bounded away from zero toward the boundary of D, and p is a holomorphic self-map of D such that the weighted composition operator W assigning the product of m and the composition of f and p to a given function f is compact on a holomorphic functional Hilbert space (containing the polynomial functions densely) on D with reproducing kernel K blowing up along the diagonal of D toward its boundary, then p has a unique fixed point in D. We apply this result by making a reasonable conjecture about the spectrum of W based on previous one-variable and multivariable results concerning compact weighted and unweighted composition operators.

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