  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.
