Coincidence and calculation of some strict \(s\)-numbers
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.
scientific article; zbMATH DE number 6033710
scientific article