A generalization of the Aleksandrov operator and adjoints of weighted composition operators
[Une généralisation de l’opérateur d’Aleksandrov, et les adjoints des opérateurs de composition à poids]
Annales de l'Institut Fourier, Tome 63 (2013) no. 2, pp. 373-389.

On introduit une généralisation de l’opérateur d’Aleksandrov, afin de représenter l’adjoint d’un opérateur de composition à poids sur 2 par une intégrale selon une mesure. En particulier, nous montrons l’existence d’une famille de mesures qui représentent l’adjoint d’un opérateur de composition à poids, sous des hypothèses assez faibles. On discute l’unicité, et aussi la généralisation des mesures d’Aleksandrov–Clark, qui correspond au cas sans poids, c’est-à-dire au cas de l’adjoint des opérateurs de composition.

A generalization of the Aleksandrov operator is provided, in order to represent the adjoint of a weighted composition operator on 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.

DOI : 10.5802/aif.2763
Classification : 47B33, 30D55
Keywords: Aleksandrov operator, Aleksandrov–Clark measures, Weighted composition operators
Mot clés : Opérateur d’Aleksandrov, Mesures d’Aleksandrov–Clark, Opérateur de composition à poids
Gallardo-Gutiérrez, Eva A. 1 ; Partington, Jonathan R. 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.)
@article{AIF_2013__63_2_373_0,
     author = {Gallardo-Guti\'errez, Eva A. and Partington, Jonathan R.},
     title = {A generalization of the {Aleksandrov} operator and adjoints of weighted composition operators},
     journal = {Annales de l'Institut Fourier},
     pages = {373--389},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {63},
     number = {2},
     year = {2013},
     doi = {10.5802/aif.2763},
     zbl = {1282.47032},
     mrnumber = {3112515},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2763/}
}
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
SP  - 373
EP  - 389
VL  - 63
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2763/
DO  - 10.5802/aif.2763
LA  - en
ID  - AIF_2013__63_2_373_0
ER  - 
%0 Journal Article
%A Gallardo-Gutiérrez, Eva A.
%A Partington, Jonathan R.
%T A generalization of the Aleksandrov operator and adjoints of weighted composition operators
%J Annales de l'Institut Fourier
%D 2013
%P 373-389
%V 63
%N 2
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.2763/
%R 10.5802/aif.2763
%G en
%F AIF_2013__63_2_373_0
Gallardo-Gutiérrez, Eva A.; Partington, Jonathan R. A generalization of the Aleksandrov operator and adjoints of weighted composition operators. Annales de l'Institut Fourier, Tome 63 (2013) no. 2, pp. 373-389. doi : 10.5802/aif.2763. http://archive.numdam.org/articles/10.5802/aif.2763/

[1] Aleksandrov, A. B. Multiplicity of boundary values of inner functions, Izv. Akad. Nauk Armyan. SSR Ser. Mat., Volume 22 (1987) no. 5, pp. 490-503 | MR | Zbl

[2] Cima, Joseph A.; Matheson, Alec Cauchy transforms and composition operators, Illinois J. Math., Volume 42 (1998) no. 1, pp. 58-69 http://projecteuclid.org/getRecord?id=euclid.ijm/1255985613 | MR | Zbl

[3] Cima, Joseph A.; Matheson, Alec L.; Ross, William T. The Cauchy transform, Mathematical Surveys and Monographs, 125, American Mathematical Society, Providence, RI, 2006 | MR | Zbl

[4] Cowen, Carl C. Linear fractional composition operators on H 2 , Integral Equations Operator Theory, Volume 11 (1988) no. 2, pp. 151-160 | DOI | MR | Zbl

[5] Čučković, Željko; Zhao, Ruhan Weighted composition operators between different weighted Bergman spaces and different Hardy spaces, Illinois J. Math., Volume 51 (2007) no. 2, p. 479-498 (electronic) http://projecteuclid.org/getRecord?id=euclid.ijm/1258138425 | MR | Zbl

[6] Eveson, S. P. Compactness criteria for integral operators in L and L 1 spaces, Proc. Amer. Math. Soc., Volume 123 (1995) no. 12, pp. 3709-3716 | DOI | MR | Zbl

[7] Gallardo-Gutiérrez, Eva A.; González, María J.; Nieminen, Pekka J.; Saksman, Eero On the connected component of compact composition operators on the Hardy space, Adv. Math., Volume 219 (2008) no. 3, pp. 986-1001 | DOI | MR | Zbl

[8] Garnett, John B. Bounded analytic functions, Graduate Texts in Mathematics, 236, Springer, New York, 2007 | MR | Zbl

[9] Littlewood, J. E. On Inequalities in the Theory of Functions, Proc. London Math. Soc., Volume S2-23 (1925) no. 1, pp. 481-519 | DOI | MR

[10] Matheson, Alec; Stessin, Michael Applications of spectral measures, Recent advances in operator-related function theory (Contemp. Math.), Volume 393, Amer. Math. Soc., Providence, RI, 2006, pp. 15-27 | DOI | MR | Zbl

[11] Nevanlinna, R. Remarques sur le lemme de Schwarz, Comptes Rendus Acad. Sci. Paris, Volume 188 (1929), pp. 1027-1029

[12] Nieminen, Pekka J.; Saksman, Eero Boundary correspondence of Nevanlinna counting functions for self-maps of the unit disc, Trans. Amer. Math. Soc., Volume 356 (2004) no. 8, p. 3167-3187 (electronic) | DOI | MR | Zbl

[13] Nieminen, Pekka J.; Saksman, Eero On compactness of the difference of composition operators, J. Math. Anal. Appl., Volume 298 (2004) no. 2, pp. 501-522 | DOI | MR | Zbl

[14] Poltoratski, Alexei; Sarason, Donald Aleksandrov-Clark measures, Recent advances in operator-related function theory (Contemp. Math.), Volume 393, Amer. Math. Soc., Providence, RI, 2006, pp. 1-14 | DOI | MR | Zbl

[15] Saksman, Eero An elementary introduction to Clark measures, Topics in complex analysis and operator theory, Univ. Málaga, Málaga, 2007, pp. 85-136 | MR | Zbl

[16] Sarason, Donald Composition operators as integral operators, Analysis and partial differential equations (Lecture Notes in Pure and Appl. Math.), Volume 122, Dekker, New York, 1990, pp. 545-565 | MR | Zbl

[17] Shapiro, Joël H.; Sundberg, Carl Compact composition operators on L 1 , Proc. Amer. Math. Soc., Volume 108 (1990) no. 2, pp. 443-449 | DOI | MR | Zbl

Cité par Sources :