The spectrum of some Hardy kernel matrices
Annales de l'Institut Fourier, Online first, 34 p.

For α>0 we consider the operator K α : 2 2 corresponding to the matrix

(nm) -1 2+α [max(n,m)] 2α n,m=1 .

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 2 .

Pour α>0 on considère l’opérateur K α : 2 2 correspondant à la matrice

(nm) -1 2+α [max(n,m)] 2α n,m=1 .

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 2 des séries de Dirichlet.

Online First:
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
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)
     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},
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  - 2023
PB  - Association des Annales de l’institut Fourier
N1  - Online first
DO  - 10.5802/aif.3589
LA  - en
ID  - AIF_0__0_0_A55_0
ER  - 
%0 Unpublished Work
%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 2023
%I Association des Annales de l’institut Fourier
%Z Online first
%R 10.5802/aif.3589
%G en
%F AIF_0__0_0_A55_0
Brevig, Ole Fredrik; Perfekt, Karl-Mikael; Pushnitski, Alexander. The spectrum of some Hardy kernel matrices. Annales de l'Institut Fourier, Online first, 34 p.

[1] Appel, Matthew J.; Bourdon, Paul S.; Thrall, John J. Norms of composition operators on the Hardy space, Experiment. Math., Volume 5 (1996) no. 2, pp. 111-117 | DOI | MR | Zbl

[2] Brevig, Ole Fredrik Sharp norm estimates for composition operators and Hilbert-type inequalities, Bull. Lond. Math. Soc., Volume 49 (2017) no. 6, pp. 965-978 | DOI | MR | Zbl

[3] Brevig, Ole Fredrik; Perfekt, Karl-Mikael; Seip, Kristian; Siskakis, Aristomenis G.; Vukotić, Dragan The multiplicative Hilbert matrix, Adv. Math., Volume 302 (2016), pp. 410-432 | DOI | MR | Zbl

[4] Gilbert, D. J.; Pearson, D. B. On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators, J. Math. Anal. Appl., Volume 128 (1987) no. 1, pp. 30-56 | DOI | MR | Zbl

[5] Gordon, Julia; Hedenmalm, Håkan The composition operators on the space of Dirichlet series with square summable coefficients, Michigan Math. J., Volume 46 (1999) no. 2, pp. 313-329 | DOI | MR | Zbl

[6] Hardy, G. H.; Littlewood, J. E.; Pólya, G. Inequalities, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988, xii+324 pages (Reprint of the 1952 edition) | MR | Zbl

[7] Hedenmalm, Håkan Dirichlet series and functional analysis, The legacy of Niels Henrik Abel, Springer, Berlin, 2004, pp. 673-684 | MR

[8] Kalvoda, Tomáš; Šťovíček, Pavel A family of explicitly diagonalizable weighted Hankel matrices generalizing the Hilbert matrix, Linear Multilinear Algebra, Volume 64 (2016) no. 5, pp. 870-884 | DOI | MR | Zbl

[9] Khan, S.; Pearson, D. B. Subordinacy and spectral theory for infinite matrices, Helv. Phys. Acta, Volume 65 (1992) no. 4, pp. 505-527 | MR

[10] Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010, xx+578 pages (With a foreword by Tom H. Koornwinder) | DOI | MR | Zbl

[11] Miheisi, Nazar; Pushnitski, Alexander A Helson matrix with explicit eigenvalue asymptotics, J. Funct. Anal., Volume 275 (2018) no. 4, pp. 967-987 | DOI | MR | Zbl

[12] Miheisi, Nazar; Pushnitski, Alexander Restriction theorems for Hankel operators, Studia Math., Volume 254 (2020) no. 1, pp. 1-21 | DOI | MR | Zbl

[13] Muthukumar, Perumal; Ponnusamy, Saminathan; Queffélec, Hervé Estimate for norm of a composition operator on the Hardy–Dirichlet space, Integral Equations Operator Theory, Volume 90 (2018) no. 1, 11, 12 pages | DOI | MR | Zbl

[14] Otte, P. Diagonalization of the Hilbert matrix, Conference talk at ICDESFA, 2005 (

[15] Perfekt, Karl-Mikael; Pushnitski, Alexander On the spectrum of the multiplicative Hilbert matrix, Ark. Mat., Volume 56 (2018) no. 1, pp. 163-183 | DOI | MR | Zbl

[16] Pushnitski, Alexander The spectral density of Hardy kernel matrices (to appear in J. Operator Theory,

[17] Queffélec, Hervé; Queffélec, Martine Diophantine approximation and Dirichlet series, Harish-Chandra Research Institute Lecture Notes, 2, Hindustan Book Agency, New Delhi, 2013, xii+232 pages | DOI | MR | Zbl

[18] Rosenblum, Marvin On the Hilbert matrix. II, Proc. Amer. Math. Soc., Volume 9 (1958), pp. 581-585 | DOI | MR | Zbl

[19] Schmüdgen, Konrad Unbounded self-adjoint operators on Hilbert space, Graduate Texts in Mathematics, 265, Springer, Dordrecht, 2012, xx+432 pages | DOI | MR | Zbl

[20] Schur, J. Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen, J. Reine Angew. Math., Volume 140 (1911), pp. 1-28 | DOI | MR

[21] Štampach, František The Hilbert L-matrix, J. Funct. Anal., Volume 282 (2022) no. 8, 109401, 46 pages | DOI | MR | Zbl

[22] Teschl, Gerald Jacobi operators and completely integrable nonlinear lattices, Mathematical Surveys and Monographs, 72, American Mathematical Society, Providence, RI, 2000, xvii+351 pages | DOI | MR | Zbl

[23] Wilf, Herbert S. On Dirichlet series and Toeplitz forms, J. Math. Anal. Appl., Volume 8 (1964), pp. 45-51 | DOI | MR | Zbl

[24] Wong, R.; Li, H. Asymptotic expansions for second-order linear difference equations, J. Comput. Appl. Math., Volume 41 (1992) no. 1-2, pp. 65-94 (Asymptotic methods in analysis and combinatorics) | DOI | MR | Zbl

Cited by Sources: