Coincidence and calculation of some strict \(s\)-numbers
type: Item
instance of: scholarly article
title: Coincidence and calculation of some strict \(s\)-numbers
zbMATH Open document ID: 1257.47021
DOI: 10.4171/ZAA/1453
author: D. E. Edmunds
author: Jan Lang
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2012-05-11T00:00:00Z
review text: The paper considers the so-called strict \(s\)-numbers, which form an important subclass of the family of all \(s\)-numbers. An \(s\)-number scale \(s\) is said to be strict if \(s_n(\text{id}: E \rightarrow E)=1\) for every Banach space \(E\) with \(\dim(E) \geq n\) (rather than only for at least \(E=\ell_2^n\) as for the larger class of all \(s\)-numbers). For operators acting between Hilbert spaces, the various \(s\)-numbers are known to coincide. In the article under review, the authors give examples of linear maps \(T\) and non-Hilbert spaces \(X, Y\) such that all strict \(s\)-numbers of \(T : X \to Y\) coincide. The maps considered are either simple integral operators acting in Lebesgue spaces or Sobolev embeddings; in these cases, the exact value of the strict \(s\)-numbers is determined. The authors' arguments crucially depend on generalized trigonometric functions.
reviewed by: Carsten Michels
Mathematics Subject Classification ID: 47B06
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 46B50
zbMATH DE Number: 6033710
zbMATH Keywords: \(s\)-numbers
zbMATH Keywords: generalized trigonometric functions
zbMATH Keywords: Sobolev embedding
zbMATH Keywords: Hardy operator
zbMATH Keywords: widths
zbMATH Keywords: compact maps
zbMATH Keywords: asymptotic estimates
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.4171/zaa/1453
OpenAlex ID: W2335201595
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description: scientific article; zbMATH DE number 6033710