2012/01/24 by Patrick Gerard, Patrick Gérard, Gerard, Patrick +2
Mathematics · #Holomorphic and Operator Theory #Mathematical Analysis and Transform Methods #Spectral Theory in Mathematical Physics #math.AP #math.FA
paper · pdf · doi:10.48550/arxiv.1201.4971
25 pages
arxiv created 2012/01/24 · arxiv updated 2012/01/25
Given two arbitrary sequences (λj)j≥ 1 and (μj)j≥ 1 of real numbers satisfying |λ1|>|μ1|>|λ2|>|μ2|>...>| λj| >| μj| → 0 , we prove that there exists a unique sequence c=(cn)n∈\Z+, real valued, such that the Hankel operators Γc and Γ c of symbols c=(cn)n≥ 0 and c=(cn+1)n≥ 0 respectively, are selfadjoint compact operators on ℓ2(\Z+) and have the sequences (λj)j≥ 1 and (μj)j≥ 1 respectively as non zero eigenvalues. Moreover, we give an explicit formula for c and we describe the kernel of Γc and of Γ c in terms of the sequences (λj)j≥ 1 and (μj)j≥ 1. More generally, given two arbitrary sequences (ρj)j≥ 1 and (σj)j≥ 1 of positive numbers satisfying ρ1>σ1>ρ2>σ2>...> ρj> σj → 0 , we describe the set of sequences c=(cn)n∈\Z+ of complex numbers such that the Hankel operators Γc and Γ c are compact on ℓ 2(\Z+) and have sequences (ρj)j≥ 1 and (σj)j≥ 1 respectively as non zero singular values.