Normal weighted composition operators on weighted Dirichlet spaces

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Author(s)
Li, Lu
Nakada, Yukihide
Nestor, Douglas
Shang, Wendy
Weir, Rachel
Date Issued
March 1, 2015
Abstract
Let φ be an analytic self-map of D with φ ( p ) = p for some p ∈ D , let ψ be bounded and analytic on D , and consider the weighted composition operator W ψ , φ defined by W ψ , φ f = ψ ⋅ ( f ∘ φ ) . On the Hardy space and Bergman space, it is known that W ψ , φ is bounded and normal precisely when ψ = c K p / ( K p ∘ φ ) and φ = α p ∘ ( δ α p ) , where K p is the reproducing kernel for the space, α p ( z ) = ( p − z ) / ( 1 − p ¯ z ) , and δ and c are constants with | δ | ≤ 1 . In particular, in this setting, φ is necessarily linear-fractional. Motivated by this result, we characterize the bounded, normal weighted composition operators W ψ , φ on the Dirichlet space D in the case when φ is linear-fractional with fixed point p ∈ D , showing that no nontrivial normal weighted composition operators of this form exist on D . Our methods also allow us to extend this result to certain weighted Dirichlet spaces in the case when φ is not an automorphism.
Description
This article will not be available on DSpace. The publisher does not allow Allegheny College to post this article on DSpace due to College’s OA mandate.
Journal
Journal of Mathematical Analysis and Applications
Department
Mathematics
Citation
Li, Lu, et al. 2015. "Normal weighted composition operators on weighted Dirichlet spaces.” Journal of Mathematical Analysis and Applications 423, no. 1: 758-769.
Publisher
Elsevier
DOI
10.1016/j.jmaa.2014.10.011
ISSN
0022-247X
1096-0813
Subjects

Dirichlet space

Composition operators...

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