Trace et valeurs propres extrêmes d'un produit de matrices de Toeplitz. Le cas singulier

2011/06/10 by Philippe Rambour, Rambour, Philippe, Abdellatif Seghier +1
Mathematics · #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Mathematical Dynamics and Fractals #Operator Algebras (math.OA) #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.1106.2072

openalex publication_date 2011/06/10 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

Trace and extreme eigenvalues of a product of truncated Toeplitz matrices. The singular case. In a first theorem we give an asymptotic expansion of Tr (T_N (f₁) T_N^-1(f₂)) where f1 (θ) = |1 - e^{i θ} | ^2α1c1 (eiθθ) and f2 (θ) = |1 - e ^iθ| ^2α2c2 (eiθ), with c1 and c2 are two regular functions of the torus and - 1/2 < α1, α2 < 1/2 . In a second part of this work we study the particular case where α1 > 0 and α2 < 0. Then we obtain the asymptotic of the trace of the powers of Tr (T_N (f₁) T_N^-1(f₂)) for s ∈ N* that provides us the limits when N goes to the infinity of the extreme eigenvalues of this matrix. This last result allows us to give a large deviation principle for a family of quadratic forms of stationnary process.

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