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

Gallardo-Gutiérrez, Eva A.; Partington, Jonathan R.

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 ${\mathscr{H}}^2$ 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 ${\mathscr{H}}^2$ 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.

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