Compact differences of composition operators on weighted Dirichlet spaces

Abstract

Here we consider when the difference of two composition operators is compact on the weighted Dirichlet spaces D α . Specifically we study differences of composition operators on the Dirichlet space D and S 2 , the space of analytic functions whose first derivative is in H 2 , and then use Calderón's complex interpolation to extend the results to the general weighted Dirichlet spaces. As a corollary we consider composition operators induced by linear fractional self-maps of the disk. MSC:47B33, 46E20, 47B32

Topological Structures of Derivative Weighted Composition Operators on the Bergman Space

“…Our discussion begins with the characterization of the compact difference of -composition operators. When we accomplished the following theorem, we read a new paper [22], in which Allen et al obtained the same estimation (Lemma 10). Nonetheless, we present our version in the following, since our method is totally different.…”

Compact differences of weighted composition operators on the weighted Bergman spaces

“…While the same question was answered positively on the weighted Bergman spaces [13], this turned out to be not true on the Hardy space [14]. Some other results on differences of weighted composition operators on spaces of holomorphic functions can be found, for example, in [15–21]. In relation with the study of the topological structures, the difference or the linear sum of composition operators on various settings has been a very active topic [13, 17, 22–24].…”