Hermitian Weighted Composition Operators and Bergman Extremal Functions

Carl C. Cowen, Gajath Gunatillake, Eungil Ko

Research output: Contribution to journalArticlepeer-review

17 Scopus citations

Abstract

Weighted composition operators have been related to products of composition operators and their adjoints and to isometries of Hardy spaces. In this paper, Hermitian weighted composition operators on weighted Hardy spaces of the unit disk are studied. In particular, necessary conditions are provided for a weighted composition operator to be Hermitian on such spaces. On weighted Hardy spaces for which the kernel functions are (1 - w̄z) for κ ≥ 1, including the standard weight Bergman spaces, the Hermitian weighted composition operators are explicitly identified and their spectra and spectral decompositions are described. Some of these Hermitian operators are part of a family of closely related normal weighted composition operators. In addition, as a consequence of the properties of weighted composition operators, we compute the extremal functions for the subspaces associated with the usual atomic inner functions for these weighted Bergman spaces and we also get explicit formulas for the projections of the kernel functions on these subspaces.

Original languageEnglish
Pages (from-to)69-99
Number of pages31
JournalComplex Analysis and Operator Theory
Volume7
Issue number1
DOIs
StatePublished - Feb 2013

Keywords

  • Bergman inner function
  • Composition operator
  • Hermitian operator
  • Normal operator
  • Projected kernel function
  • Spectral measure
  • Weighted Bergman space
  • Weighted composition operator
  • Weighted Hardy space

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