For we consider the operator corresponding to the matrix
By interpreting as the inverse of an unbounded Jacobi matrix, we show that the absolutely continuous spectrum coincides with (multiplicity one), and that there is no singular continuous spectrum. There is a finite number of eigenvalues above the continuous spectrum. We apply our results to demonstrate that the reproducing kernel thesis does not hold for composition operators on the Hardy space of Dirichlet series .
Pour on considère l’opérateur correspondant à la matrice
En interprétant comme l’inverse d’une matrice de Jacobi non bornée, on montre que le spectre absolument continu coïncide avec (multiplicité un), et qu’il n’y a pas de spectre continu singulier. Il existe un nombre fini de valeurs propres au-dessus du spectre continu. Nous appliquons nos résultats pour démontrer que la thèse du noyau reproduisant est en défaut pour les opérateurs de composition sur l’espace de Hardy des séries de Dirichlet.
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Keywords: Spectrum, Jacobi matrix, Hardy kernel, Hardy space of Dirichlet series, composition operator
Mot clés : Spectre, matrice de Jacobi, noyau de Hardy, espace de Hardy de séries de Dirichlet, opérateur de composition
Brevig, Ole Fredrik 1; Perfekt, Karl-Mikael 2; Pushnitski, Alexander 3
@article{AIF_2024__74_3_1061_0, author = {Brevig, Ole Fredrik and Perfekt, Karl-Mikael and Pushnitski, Alexander}, title = {The spectrum of some {Hardy} kernel matrices}, journal = {Annales de l'Institut Fourier}, pages = {1061--1094}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {74}, number = {3}, year = {2024}, doi = {10.5802/aif.3589}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3589/} }
TY - JOUR AU - Brevig, Ole Fredrik AU - Perfekt, Karl-Mikael AU - Pushnitski, Alexander TI - The spectrum of some Hardy kernel matrices JO - Annales de l'Institut Fourier PY - 2024 SP - 1061 EP - 1094 VL - 74 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3589/ DO - 10.5802/aif.3589 LA - en ID - AIF_2024__74_3_1061_0 ER -
%0 Journal Article %A Brevig, Ole Fredrik %A Perfekt, Karl-Mikael %A Pushnitski, Alexander %T The spectrum of some Hardy kernel matrices %J Annales de l'Institut Fourier %D 2024 %P 1061-1094 %V 74 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3589/ %R 10.5802/aif.3589 %G en %F AIF_2024__74_3_1061_0
Brevig, Ole Fredrik; Perfekt, Karl-Mikael; Pushnitski, Alexander. The spectrum of some Hardy kernel matrices. Annales de l'Institut Fourier, Volume 74 (2024) no. 3, pp. 1061-1094. doi : 10.5802/aif.3589. https://aif.centre-mersenne.org/articles/10.5802/aif.3589/
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