New criteria for boundedness and compactness of weighted composition operators mapping into the Bloch space

Let ψ and φ be analytic functions on the open unit disk with φ( ) ⊆ . We give new characterizations of the bounded and compact weighted composition operators W ψ,ϕ from the Hardy spaces H p , 1 ≤ p ≤ ∞, the Bloch space B , the weighted Bergman spaces A α p , α > − 1,1 ≤ p < ∞, and the Dirichle...

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Bibliographic Details
Published in:Central European journal of mathematics 2013-01, Vol.11 (1), p.55-73
Main Author: Colonna, Flavia
Format: Article
Language:English
Subjects:
30H10
30H20
30H30
47B38
Algebra
Analytic functions
Bloch space
Composition
Dirichlet problem
Dirichlet space
Geometry
Hardy space
Lie Groups
Mathematics
Mathematics and Statistics
Norms
Number Theory
Operators (mathematics)
Probability Theory and Stochastic Processes
Research Article
Topological Groups
Weighted Bergman space
Weighted composition operators
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Summary:Let ψ and φ be analytic functions on the open unit disk with φ( ) ⊆ . We give new characterizations of the bounded and compact weighted composition operators W ψ,ϕ from the Hardy spaces H p , 1 ≤ p ≤ ∞, the Bloch space B , the weighted Bergman spaces A α p , α > − 1,1 ≤ p < ∞, and the Dirichlet space to the Bloch space in terms of boundedness (respectively, convergence to 0) of the Bloch norms of W ψ,ϕ f for suitable collections of functions f in the respective spaces. We also obtain characterizations of boundedness for H 1 as well as of compactness for H p , 1 ≤ p < ∞, and purely in terms of the symbols ψ and φ .
ISSN:1895-1074
1644-3616
2391-5455
DOI:10.2478/s11533-012-0097-4