A generalization of the Aleksandrov operator and adjoints of weighted composition operators
Annales de l'Institut Fourier, Volume 63 (2013) no. 2, p. 373-389
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.
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.
DOI : https://doi.org/10.5802/aif.2763
Classification:  47B33,  30D55
Keywords: Aleksandrov operator, Aleksandrov–Clark measures, Weighted composition operators
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {63},
     number = {2},
     year = {2013},
     pages = {373-389},
     doi = {10.5802/aif.2763},
     mrnumber = {3112515},
     zbl = {1282.47032},
     language = {en},
     url = {http://www.numdam.org/item/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, Volume 63 (2013) no. 2, pp. 373-389. doi : 10.5802/aif.2763. http://www.numdam.org/item/AIF_2013__63_2_373_0/

[1] Aleksandrov, A. B. Multiplicity of boundary values of inner functions, Izv. Akad. Nauk Armyan. SSR Ser. Mat., Tome 22 (1987) no. 5, pp. 490-503 | MR 931885 | Zbl 0648.30002

[2] Cima, Joseph A.; Matheson, Alec Cauchy transforms and composition operators, Illinois J. Math., Tome 42 (1998) no. 1, pp. 58-69 http://projecteuclid.org/getRecord?id=euclid.ijm/1255985613 | MR 1492039 | Zbl 0914.30023

[3] Cima, Joseph A.; Matheson, Alec L.; Ross, William T. The Cauchy transform, American Mathematical Society, Providence, RI, Mathematical Surveys and Monographs, Tome 125 (2006) | MR 2215991 | Zbl 1096.30046

[4] Cowen, Carl C. Linear fractional composition operators on H 2 , Integral Equations Operator Theory, Tome 11 (1988) no. 2, pp. 151-160 | Article | MR 928479 | Zbl 0638.47027

[5] Čučković, Željko; Zhao, Ruhan Weighted composition operators between different weighted Bergman spaces and different Hardy spaces, Illinois J. Math., Tome 51 (2007) no. 2, p. 479-498 (electronic) http://projecteuclid.org/getRecord?id=euclid.ijm/1258138425 | MR 2342670 | Zbl 1147.47021

[6] Eveson, S. P. Compactness criteria for integral operators in L and L 1 spaces, Proc. Amer. Math. Soc., Tome 123 (1995) no. 12, pp. 3709-3716 | Article | MR 1291766 | Zbl 0841.47028

[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., Tome 219 (2008) no. 3, pp. 986-1001 | Article | MR 2442059 | Zbl 1187.47021

[8] Garnett, John B. Bounded analytic functions, Springer, New York, Graduate Texts in Mathematics, Tome 236 (2007) | MR 2261424 | Zbl 1106.30001

[9] Littlewood, J. E. On Inequalities in the Theory of Functions, Proc. London Math. Soc., Tome S2-23 (1925) no. 1, pp. 481-519 | Article | MR 1575208

[10] Matheson, Alec; Stessin, Michael Applications of spectral measures, Recent advances in operator-related function theory, Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 393 (2006), pp. 15-27 | Article | MR 2198104 | Zbl 1116.47020

[11] Nevanlinna, R. Remarques sur le lemme de Schwarz, Comptes Rendus Acad. Sci. Paris, Tome 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., Tome 356 (2004) no. 8, p. 3167-3187 (electronic) | Article | MR 2052945 | Zbl 1210.30012

[13] Nieminen, Pekka J.; Saksman, Eero On compactness of the difference of composition operators, J. Math. Anal. Appl., Tome 298 (2004) no. 2, pp. 501-522 | Article | MR 2086972 | Zbl 1072.47021

[14] Poltoratski, Alexei; Sarason, Donald Aleksandrov-Clark measures, Recent advances in operator-related function theory, Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 393 (2006), pp. 1-14 | Article | MR 2198367 | Zbl 1102.30032

[15] Saksman, Eero An elementary introduction to Clark measures, Topics in complex analysis and operator theory, Univ. Málaga, Málaga (2007), pp. 85-136 | MR 2394657 | Zbl 1148.47001

[16] Sarason, Donald Composition operators as integral operators, Analysis and partial differential equations, Dekker, New York (Lecture Notes in Pure and Appl. Math.) Tome 122 (1990), pp. 545-565 | MR 1044808 | Zbl 0712.47026

[17] Shapiro, Joël H.; Sundberg, Carl Compact composition operators on L 1 , Proc. Amer. Math. Soc., Tome 108 (1990) no. 2, pp. 443-449 | Article | MR 994787 | Zbl 0704.47018