The spectrum of some Hardy kernel matrices

Brevig, Ole Fredrik; Perfekt, Karl-Mikael; Pushnitski, Alexander

doi:10.5802/aif.3589

The spectrum of some Hardy kernel matrices
[Le spectre de certaines matrices du noyau de Hardy]

Brevig, Ole Fredrik ¹ ; Perfekt, Karl-Mikael ² ; Pushnitski, Alexander ³

¹ Department of Mathematics University of Oslo 0851 Oslo (Norway)
² Department of Mathematical Sciences Norwegian University of Science and Technology (NTNU) 7491 Trondheim (Norway)
³ Department of Mathematics King’s College London Strand, London WC2R 2LS (United Kingdom)

Annales de l'Institut Fourier, Online first, 34 p.

Résumé
Abstract

Pour $α > 0$ on considère l’opérateur $K_{α} : ℓ^{2} \to ℓ^{2}$ correspondant à la matrice

{(\frac{{(n m)}^{- \frac{1}{2} + α}}{{[max (n, m)]}^{2 α}})}_{n, m = 1}^{\infty} .

En interprétant $K_{α}$ comme l’inverse d’une matrice de Jacobi non bornée, on montre que le spectre absolument continu coïncide avec $[0, 2 / α]$ (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 $ℋ^{2}$ des séries de Dirichlet.

For $α > 0$ we consider the operator $K_{α} : ℓ^{2} \to ℓ^{2}$ corresponding to the matrix

{(\frac{{(n m)}^{- \frac{1}{2} + α}}{{[max (n, m)]}^{2 α}})}_{n, m = 1}^{\infty} .

By interpreting $K_{α}$ as the inverse of an unbounded Jacobi matrix, we show that the absolutely continuous spectrum coincides with $[0, 2 / α]$ (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 $ℋ^{2}$ .

Reçu le : 2020-04-20
Révisé le : 2022-04-01
Accepté le : 2022-05-12
Première publication : 2023-07-03

DOI : 10.5802/aif.3589

Classification : 47A10, 47B33, 47B34, 47B36
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

Affiliations des auteurs :

Brevig, Ole Fredrik ¹ ; Perfekt, Karl-Mikael ² ; Pushnitski, Alexander ³

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     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},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2023},
     doi = {10.5802/aif.3589},
     language = {en},
     note = {Online first},
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Brevig, Ole Fredrik; Perfekt, Karl-Mikael; Pushnitski, Alexander. The spectrum of some Hardy kernel matrices. Annales de l'Institut Fourier, Online first, 34 p.

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