Necessary condition for compactness of a difference of composition operators on the Dirichlet space

Abstract

Let ϕ be a self-map of the unit disk and let C ϕ denote the composition operator acting on the standard Dirichlet space D. A necessary condition for compactness of a difference of two bounded composition operators acting on D, is given. As an application, a characterization of disk automorphisms ϕ and ψ for which the commutatorwhere the branch of the logarithm is chosen such thatBy a self-map of D we mean an analytic function ϕ such that ϕ(D) ⊂ D. We will also assume that a self-map ϕ is not a constant function. For a self-map of the unit disk ϕ, the composition operator C ϕ on the Dirichlet space D is defined by C ϕ f := f • ϕ. The composition operator C ϕ on Dirichlet space is not necessarily bounded for an arbitrary self-map of the unit disk. However, C ϕ is bounded on D if, for example, ϕ is a finitely Date: May 22, 2018. 2010 Mathematics Subject Classification. 47B33.

Compact differences of weighted composition operators on the weighted Bergman spaces

“…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]. Recently, Moorhouse [1] characterized completely the compactness for the difference of two composition operators on the Bergman space over the unit disk, and Al-Rawashdeh and Narayan [25] gave a sufficient condition for the same problem on the Hardy space.…”