Asymptotic behaviour of eigenvalues of Hankel operators

2014/12/08 by Pushnitski, Alexander, Yafaev, Dmitri
#47B06 #47B35 #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1412.2633

Abstract

We consider compact Hankel operators realized in ℓ²(\mathbb Z₊) as infinite matrices Γ with matrix elements h(j+k). Roughly speaking, we show that if h(j)∼ (b₁+ (-1)^j b_-1) j^-1(log j)^-α as j→ ∞ for some α>0, then the eigenvalues of Γ satisfy λ_n^± (Γ)∼ c^± n^-α as n→ ∞. The asymptotic coefficients c^± are explicitly expressed in terms of the asymptotic coefficients b₁ and b_-1. Similar results are obtained for Hankel operators \mathbf Γ realized in L²(\mathbb R₊) as integral operators with kernels \mathbf h(t+s). In this case the asymptotics of eigenvalues λ_n^± (\mathbf Γ) are determined by the behaviour of \mathbf h(t) as t→ 0 and as t→ ∞.

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