Optimal Weyl-type inequalities for operators in Banach spaces
type: Item
instance of: scholarly article
title: Optimal Weyl-type inequalities for operators in Banach spaces
zbMATH Open document ID: 1123.47019
author: Bernd Carl
author: Aicke Hinrichs
published in: Positivity
publication date: 2007-05-24T00:00:00Z
review text: Let \((s_n)\) be an \(s\)-number sequence. The authors show, for each \(k = 1, 2, \dots\) and \(n \geq k + 1\), the inequality \[ {\left( \prod _{j=1}^n | \lambda_j (T)| \right)}^{1/n}\leq (k+1)^{1/2}\| T\| ^{1-k/n} {\left( \prod^n _{j=n-k+1} s_j(T)\right)}^{1/n} \] between the eigenvalues and \(s\)-numbers of a compact operator \(T\) in a Banach space. The proof is given for the smallest scale of \(s\)-numbers, namely the Hilbert numbers, for which several other inequalities are shown to this purpose. Using Fourier matrices and diagonal matrices, the authors confirm that the constant \((k + 1)^{1/2}\) is optimal for \(n = k + 1\) and \(k = 1, 2,\dots\). They also give some hints why this inequality seems to be an appropriate tool for estimating the first single eigenvalues. On the other hand, the authors prove that the Weyl numbers form a minimal multiplicative \(s\)-number sequence in the following sense. Let \((s_n)\) be a multiplicative \(s\)-number sequence. Then, if for all linear and continuous operators \(T\) between Banach spaces it holds that \(s_n(T) \leq x_n(T)\), \(n=1,2, \dots\), it follows that \(x_{2n-1}(T) \leq e \, (\prod_{k=1}^{n} s_k(T))^{1/n}\) for \(n=1,2, \dots\).
reviewed by: Carsten Michels
Mathematics Subject Classification ID: 47B06
Mathematics Subject Classification ID: 47A30
zbMATH DE Number: 5156940
zbMATH Keywords: eigenvalues
zbMATH Keywords: \(s\)-numbers
zbMATH Keywords: Weyl numbers
zbMATH Keywords: Hilbert numbers
zbMATH Keywords: inequalities
zbMATH Keywords: compact operators
MaRDI profile type: Publication
full work available at URL: s11117-006-1088-0
OpenAlex ID: W2070971536
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DOI: 10.1007/S11117-006-1088-0
description: scientific article; zbMATH DE number 5156940