Linear Combinations of Composition Operators on Weighted Dirichlet Spaces

In this paper, linear combinations of composition operators acting on weighted Dirichlet spaces are studied. By using the first derivative of the kernel function, we obtain a lower estimate for the essential norms of these operators acting on the Dirichlet space ? and 2S. For general weighted Dirichlet space, by using complex interpolation methods, we characterize the compactness of these operators induced by linear fractional self-maps of the disk.