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The spectrum of some Hardy kernel matrices

Ole Fredrik Brevig, Karl-Mikael Perfekt, Alexander Pushnitski

Abstract

For $α> 0$ we consider the operator $K_α\colon \ell^2 \to \ell^2$ corresponding to the matrix \[\left(\frac{(nm)^{-\frac{1}{2}+α}}{[\max(n,m)]^{2α}}\right)_{n,m=1}^\infty.\] 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 $\mathscr{H}^2$.

The spectrum of some Hardy kernel matrices

Abstract

For we consider the operator corresponding to the matrix \[\left(\frac{(nm)^{-\frac{1}{2}+α}}{[\max(n,m)]^{2α}}\right)_{n,m=1}^\infty.\] 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 .

Paper Structure

This paper contains 27 sections, 9 theorems, 149 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Hardy kernels
  3. The kernels $k_\alpha$
  4. Connection with Jacobi matrices
  5. Failure of the reproducing kernel thesis
  6. Structure of the paper
  7. Related work
  8. The Jacobi matrices $J_\alpha$
  9. Warm-up: the continuous analogue of $J_\alpha$
  10. Factorisation of $J_\alpha$
  11. Proof of Theorem C (i)
  12. Proof of Theorem C (ii)
  13. Proof of Theorem C (iii)
  14. Asymptotics for generalised eigenvectors
  15. Subordinacy theory and the spectrum of $J_\alpha$
  16. ...and 12 more sections

Key Result

Lemma 2.1

The matrix $J_\alpha$ can be represented as

Figures (1)

  • Figure 1: Spectrum of $\alpha K_\alpha$ for $0< \alpha < 12$. The grey area is the a.c. spectrum and the black curves are the eigenvalues.

Theorems & Definitions (23)

  • Remark
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Remark
  • Theorem 2.3
  • proof : Proof of Theorem \ref{['thm:Jspec']}
  • Lemma 3.1
  • Remark
  • proof
  • ...and 13 more