Adjoints of linear fractional composition operators on the Dirichlet space
type: Item
instance of: scholarly article
title: Adjoints of linear fractional composition operators on the Dirichlet space
zbMATH Open document ID: 1048.47016
author: Eva A. Gallardo Gutiérrez
author: Alfonso Montes-Rodriguez
published in: Mathematische Annalen
publication date: 2003-11-04T00:00:00Z
review text: Let \(T\) be an open unit disc on the complex plane and \(\mathcal{D}\) the Dirichlet space of functions analytic on \(T\) equipped with the norm defined by \[ \| f\| ^2=| f(0)| ^2+\int\limits_T| f^\prime(z)| ^2\,dA(z). \] The authors study the adjoints of the composition operators \(C_\varphi f= f\circ \varphi\) with a fractional linear function \(\varphi(z)=\frac{az+b}{cz+d}\), \(ad-bc \neq 0\), \(\varphi(T)\subset T\). It is shown that these adjoints may be expressed as a fractional linear operator plus a rank two operator acting on \(\mathcal{D}\) itself. The authors observe that the proofs are different from those known for the Hardy spaces. The key point in that difference is due to the fact that many fractional linear composition operators are normal on Dirichlet spaces while \(C_\varphi\) is normal on Hardy spaces if and only if \(\varphi(z)=\lambda z\), \(| \lambda| \leq 1\). As an application, there is given an explicit formula for the norm of a fractional linear composition operator on the Dirichlet spaces modulo constant functions. As another application, the computation of the spectrum and essential spectrum is considered.
Mathematics Subject Classification ID: 47B33
Mathematics Subject Classification ID: 46E15
zbMATH DE Number: 2001051
zbMATH Keywords: Dirichlet spaces
zbMATH Keywords: composition operator
zbMATH Keywords: fractional linear map
zbMATH Keywords: adjoint operator
zbMATH Keywords: Hardy space
reviewed by: S. G. Samko
MaRDI profile type: Publication
full work available at URL: s00208-003-0442-9
OpenAlex ID: W1976482228
cites work: Q4177796
cites work: Norms of Composition Operators on the Hardy Space
cites work: Q3223360
cites work: Q4741262
cites work: Linear fractional composition operators on \(H^ 2\)
cites work: Q4888816
cites work: The role of the spectrum in the cyclic behavior of composition operators
cites work: Q5521075
cites work: The spectra of composition operators from linear fractional maps acting upon the Dirichlet space
cites work: Relating composition operators on different weighted Hardy spaces
cites work: Composition operators on Hardy spaces of a half-plane
cites work: Q4340161
cites work: Q3293614
cites work: On norms of composition operators acting on Bergman spaces.
DOI: 10.1007/S00208-003-0442-9
description: scientific article; zbMATH DE number 2001051