# 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

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topGallardo-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

top- A. B. Aleksandrov, Multiplicity of boundary values of inner functions, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 22 (1987), 490-503 Zbl0648.30002MR931885
- Joseph A. Cima, Alec Matheson, Cauchy transforms and composition operators, Illinois J. Math. 42 (1998), 58-69 Zbl0914.30023MR1492039
- Joseph A. Cima, Alec L. Matheson, William T. Ross, The Cauchy transform, 125 (2006), American Mathematical Society, Providence, RI Zbl1096.30046MR2215991
- Carl C. Cowen, Linear fractional composition operators on ${H}^{2}$, Integral Equations Operator Theory 11 (1988), 151-160 Zbl0638.47027MR928479
- Željko Čučković, Ruhan Zhao, Weighted composition operators between different weighted Bergman spaces and different Hardy spaces, Illinois J. Math. 51 (2007), 479-498 (electronic) Zbl1147.47021MR2342670
- S. P. Eveson, Compactness criteria for integral operators in ${L}^{\infty}$ and ${L}^{1}$ spaces, Proc. Amer. Math. Soc. 123 (1995), 3709-3716 Zbl0841.47028MR1291766
- Eva A. Gallardo-Gutiérrez, María J. González, Pekka J. Nieminen, Eero Saksman, On the connected component of compact composition operators on the Hardy space, Adv. Math. 219 (2008), 986-1001 Zbl1187.47021MR2442059
- John B. Garnett, Bounded analytic functions, 236 (2007), Springer, New York Zbl1106.30001MR2261424
- J. E. Littlewood, On Inequalities in the Theory of Functions, Proc. London Math. Soc. S2-23 (1925), 481-519 Zbl51.0247.03MR1575208
- Alec Matheson, Michael Stessin, Applications of spectral measures, Recent advances in operator-related function theory 393 (2006), 15-27, Amer. Math. Soc., Providence, RI Zbl1116.47020MR2198104
- R. Nevanlinna, Remarques sur le lemme de Schwarz, Comptes Rendus Acad. Sci. Paris 188 (1929), 1027-1029
- 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
- Pekka J. Nieminen, Eero Saksman, On compactness of the difference of composition operators, J. Math. Anal. Appl. 298 (2004), 501-522 Zbl1072.47021MR2086972
- 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
- Eero Saksman, An elementary introduction to Clark measures, Topics in complex analysis and operator theory (2007), 85-136, Univ. Málaga, Málaga Zbl1148.47001MR2394657
- Donald Sarason, Composition operators as integral operators, Analysis and partial differential equations 122 (1990), 545-565, Dekker, New York Zbl0712.47026MR1044808
- 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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