Normal weighted composition operators on weighted Dirichlet spaces

Li, Lu; Nakada, Yukihide; Nestor, Douglas; Shang, Wendy; Weir, Rachel

doi:10.1016/j.jmaa.2014.10.011

Normal weighted composition operators on weighted Dirichlet spaces

Persistent URL

http://hdl.handle.net/10456/37733

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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