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Hermitian Weighted Composition Operators and Bergman Extremal Functions

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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 \({(1 - \overline{w}z)^{-\kappa}}\) 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.

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Authors and Affiliations

  1. IUPUI (Indiana University–Purdue University Indianapolis), Indianapolis, IN, 46202, USA

    Carl C. Cowen

  2. American University of Sharjah, Sharjah, United Arab Emirates

    Gajath Gunatillake

  3. Ewha Women’s University, Seoul, 120-750, Korea

    Eungil Ko

Authors
  1. Carl C. Cowen
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  2. Gajath Gunatillake
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  3. Eungil Ko
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Corresponding author

Correspondence to Carl C. Cowen.

Additional information

Communicated by Scott McCullough.

E. Ko is supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2009-0087565). The authors would like to thank Ehwa Women’s University, Seoul, and Purdue University, West Lafayette, for their generosity in hosting the authors during their collaboration.

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Cowen, C.C., Gunatillake, G. & Ko, E. Hermitian Weighted Composition Operators and Bergman Extremal Functions. Complex Anal. Oper. Theory 7, 69–99 (2013). https://doi.org/10.1007/s11785-011-0185-7

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  • DOI: https://doi.org/10.1007/s11785-011-0185-7

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