Normal weighted composition operators on weighted Dirichlet spaces

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dc.citation.epage769en_US
dc.citation.issue1en_US
dc.citation.spage758en_US
dc.citation.volume423en_US
dc.contributor.authorLi, Lu
dc.contributor.authorNakada, Yukihide
dc.contributor.authorNestor, Douglas
dc.contributor.authorShang, Wendy
dc.contributor.authorWeir, Rachel
dc.contributor.avlauthorNakada, Yukihide
dc.contributor.avlauthorNestor, Douglas
dc.contributor.avlauthorWeir, Rachel
dc.contributor.departmentMathematicsen_US
dc.date.accessioned2015-04-07T18:43:18Z
dc.date.available2015-04-07T18:43:18Z
dc.date.issued2015-03-01
dc.descriptionThis 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.en_US
dc.description.abstractLet 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.en_US
dc.description.sponsorshipThis work was supported by: Allegheny College’s Harold M. State Research Fellowship, Dr. Barbara Lotze Student-Faculty Research Fellowship Fund, and Edward David Funden_US
dc.identifier.citationLi, Lu, et al. 2015. "Normal weighted composition operators on weighted Dirichlet spaces.” Journal of Mathematical Analysis and Applications 423, no. 1: 758-769.en_US
dc.identifier.doi10.1016/j.jmaa.2014.10.011
dc.identifier.issn0022-247X
dc.identifier.issn1096-0813
dc.identifier.urihttp://hdl.handle.net/10456/37733
dc.language.isoen_USen_US
dc.publisherElsevieren_US
dc.relation.ispartofJournal of Mathematical Analysis and Applicationsen_US
dc.relation.isversionofhttp://dx.doi.org/10.1016/j.jmaa.2014.10.011en_US
dc.subjectDirichlet spaceen_US
dc.subjectComposition operatorsen_US
dc.titleNormal weighted composition operators on weighted Dirichlet spacesen_US
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