Eigenvalue asymptotics of B(τ,ρ)

Determine the asymptotic behavior, as n → ∞, of the eigenvalues of the operator B(τ,ρ) acting on ℓ²(N) with matrix entries [B(τ,ρ)]_{n,m} = (n ∨ m)^{-τ} (nm)^{-ρ/2} in the regime τ > 0 and ρ > 1 where B(τ,ρ) is positive semi-definite and trace class.

Background

The family B(τ,ρ) consists of semi-infinite matrices acting on ℓ²(N) with entries given by (n ∨ m)^{-τ}(nm)^{-ρ/2}. For τ > 0, these matrices are positive semi-definite; when ρ > 1 they are bounded, compact, and trace class; at ρ = 1 they are bounded but non-compact with absolutely continuous spectrum; and for ρ < 1 they are unbounded.

Within this spectral overview, the paper notes that, unlike matrices A(τ,ρ) and C(τ,ρ) for which eigenvalue asymptotics are established, the asymptotics for the eigenvalues of B(τ,ρ) have not been determined. This is highlighted specifically in the trace-class case (τ > 0, ρ > 1), where such asymptotics are most naturally posed and remain unresolved.