Weighted composition operators on the Dirichlet space: boundedness and spectral properties

2015/03/03 by Chalendar, I., Gallardo-Gutiérrez, E. A., Partington, J. R.
#47B38 #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1503.01130

Abstract

Boundedness of weighted composition operators W_u,φ acting on the classical Dirichlet space D as W_u,φf= u (f∘ φ) is studied in terms of the multiplier space associated to the symbol φ, i.e., M(ϕ)=\ u ∈ \mathcal D: W_u,ϕ \hbox is bounded on \mathcal D \. A prominent role is played by the multipliers of the Dirichlet space. As a consequence, the spectrum of W_u,φ in D whenever φ is an automorphism of the unit disc is studied, extending a recent work of Hyvärinen, Lindström, Nieminen and Saukko to the context of the Dirichlet space.

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