A generalization of the Aleksandrov operator and adjoints of weighted composition operators

Eva A. Gallardo-Gutiérrez[1]; Jonathan R. Partington[2]

  • [1] Universidad Complutense de Madrid e IUMA Facultad de Ciencias Matemáticas Departamento de Análisis Matemático Plaza de Ciencias 3 28040 Madrid (Spain)
  • [2] University of Leeds School of Mathematics Leeds LS2 9JT, (U.K.)

Annales de l’institut Fourier (2013)

  • Volume: 63, Issue: 2, page 373-389
  • ISSN: 0373-0956

Abstract

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A generalization of the Aleksandrov operator is provided, in order to represent the adjoint of a weighted composition operator on ℋ 2 by means of an integral with respect to a measure. In particular, we show the existence of a family of measures which represents the adjoint of a weighted composition operator under fairly mild assumptions, and we discuss not only uniqueness but also the generalization of Aleksandrov–Clark measures which corresponds to the unweighted case, that is, to the adjoint of composition operators.

How to cite

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Gallardo-Gutiérrez, Eva A., and Partington, Jonathan R.. "A generalization of the Aleksandrov operator and adjoints of weighted composition operators." Annales de l’institut Fourier 63.2 (2013): 373-389. <http://eudml.org/doc/275503>.

@article{Gallardo2013,
abstract = {A generalization of the Aleksandrov operator is provided, in order to represent the adjoint of a weighted composition operator on $\mathcal\{H\}^2$ by means of an integral with respect to a measure. In particular, we show the existence of a family of measures which represents the adjoint of a weighted composition operator under fairly mild assumptions, and we discuss not only uniqueness but also the generalization of Aleksandrov–Clark measures which corresponds to the unweighted case, that is, to the adjoint of composition operators.},
affiliation = {Universidad Complutense de Madrid e IUMA Facultad de Ciencias Matemáticas Departamento de Análisis Matemático Plaza de Ciencias 3 28040 Madrid (Spain); University of Leeds School of Mathematics Leeds LS2 9JT, (U.K.)},
author = {Gallardo-Gutiérrez, Eva A., Partington, Jonathan R.},
journal = {Annales de l’institut Fourier},
keywords = {Aleksandrov operator; Aleksandrov–Clark measures; Weighted composition operators; Aleksandrov-Clark measures; Hardy spaces; weighted composition operators},
language = {eng},
number = {2},
pages = {373-389},
publisher = {Association des Annales de l’institut Fourier},
title = {A generalization of the Aleksandrov operator and adjoints of weighted composition operators},
url = {http://eudml.org/doc/275503},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Gallardo-Gutiérrez, Eva A.
AU - Partington, Jonathan R.
TI - A generalization of the Aleksandrov operator and adjoints of weighted composition operators
JO - Annales de l’institut Fourier
PY - 2013
PB - Association des Annales de l’institut Fourier
VL - 63
IS - 2
SP - 373
EP - 389
AB - A generalization of the Aleksandrov operator is provided, in order to represent the adjoint of a weighted composition operator on $\mathcal{H}^2$ by means of an integral with respect to a measure. In particular, we show the existence of a family of measures which represents the adjoint of a weighted composition operator under fairly mild assumptions, and we discuss not only uniqueness but also the generalization of Aleksandrov–Clark measures which corresponds to the unweighted case, that is, to the adjoint of composition operators.
LA - eng
KW - Aleksandrov operator; Aleksandrov–Clark measures; Weighted composition operators; Aleksandrov-Clark measures; Hardy spaces; weighted composition operators
UR - http://eudml.org/doc/275503
ER -

References

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  11. R. Nevanlinna, Remarques sur le lemme de Schwarz, Comptes Rendus Acad. Sci. Paris 188 (1929), 1027-1029 
  12. Pekka J. Nieminen, Eero Saksman, Boundary correspondence of Nevanlinna counting functions for self-maps of the unit disc, Trans. Amer. Math. Soc. 356 (2004), 3167-3187 (electronic) Zbl1210.30012MR2052945
  13. Pekka J. Nieminen, Eero Saksman, On compactness of the difference of composition operators, J. Math. Anal. Appl. 298 (2004), 501-522 Zbl1072.47021MR2086972
  14. Alexei Poltoratski, Donald Sarason, Aleksandrov-Clark measures, Recent advances in operator-related function theory 393 (2006), 1-14, Amer. Math. Soc., Providence, RI Zbl1102.30032MR2198367
  15. Eero Saksman, An elementary introduction to Clark measures, Topics in complex analysis and operator theory (2007), 85-136, Univ. Málaga, Málaga Zbl1148.47001MR2394657
  16. Donald Sarason, Composition operators as integral operators, Analysis and partial differential equations 122 (1990), 545-565, Dekker, New York Zbl0712.47026MR1044808
  17. Joël H. Shapiro, Carl Sundberg, Compact composition operators on L 1 , Proc. Amer. Math. Soc. 108 (1990), 443-449 Zbl0704.47018MR994787

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