Compact weighted composition operators and fixed points in convex domains
摘要
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.
引用
@article{arxiv.math/0512044,
title = {Compact weighted composition operators and fixed points in convex domains},
author = {Dana D. Clahane},
journal= {arXiv preprint arXiv:math/0512044},
year = {2007}
}
备注
10 pages. Corrected a few typographical errors and an error in one step of the main result's proof. This paper was presented in September 2005 at the Wabash Extramural Modern Analysis Mini-conference in Indianapolis