2015
DOI: 10.1016/j.jmaa.2015.02.002
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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 functio… Show more

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“…Some other results on differences of weighted composition operators on spaces of holomorphic functions can be found, for example, in [1521]. 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, 2224]. 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.…”
Section: Introductionmentioning
confidence: 99%
“…Some other results on differences of weighted composition operators on spaces of holomorphic functions can be found, for example, in [1521]. 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, 2224]. 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.…”
Section: Introductionmentioning
confidence: 99%