The spectrum of some Hardy kernel matrices

Abstract

For α > 0 we consider the operator Kα : ℓ 2 → ℓ 2 corresponding to the matrix.By interpreting Kα 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 are 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 .

The spectral density of Hardy kernel matrices

“…The spectral density of K(ϕ α ) was not computed in [3], and Theorem 2.1 in this case seems to be new. The author is grateful to Uzy Smilansky for asking the question about spectral density in this context.…”

“…The operator K(ϕ α ) was introduced in [2] in connection with a question about composition operators on the Hardy space of Dirichlet series. Some estimates for the norm of K(ϕ α ) were given in [2], and a detailed spectral analysis of this operator was accomplished in [3]. It was established that K(ϕ α ) has a.c. spectrum [0, 2/α] of multiplicity one, no singular continuous spectrum and finitely many eigenvalues above 2/α, satisfying (2.4).…”

“…This example is very special because, as shown in [3], the operator K(ϕ α ) is an inverse of a Jacobi matrix. In particular, since the spectrum of any Jacobi matrix is simple, all eigenvalues of K(ϕ α ) are simple.…”

Spectral asymptotics for Toeplitz operators and an application to banded matrices

“…One of the early appearances of these matrices in the literature can be dated back to Choi's paper [6], where the L-matrix corresponding to a n = 1/(n + 1) was called the loyal companion of Hilbert's matrix. Recently, L-matrices and their modifications started attracting some attention, beginning from the paper [4] by Bouthat and Mashreghi; see [19,20,5,2]. One of the key reasons for this was identified by Štampach, who observed in [19] that any non-singular (i.e.…”

“…Similarly, B(τ, ρ) is a weighted L-matrix with the power weight n (τ −ρ)/2 , and this structure, together with the homogeneity (1.5), makes it quite special. Particular cases of B(τ, ρ) have been studied in the already mentioned papers [4,19,5,2]. The homogeneity of B tempts one to compare its properties with the properties of the integral operator with the kernel function defined by the the same formula, see the discussion in [2,16,5].…”