Entropy numbers and interpolation

DE Edmunds, Y Netrusov

Research output: Contribution to journalArticle (Academic Journal)

13 Citations (Scopus)

Abstract

This paper settles a long-standing question by showing that in certain circumstances the entropy numbers of a map do not behave well under real interpolation. To do this, lemmas of combinatorial type are established and used to obtain lower bounds for the entropy numbers of a particular diagonal map acting between Lorentz sequence spaces. These lower bounds contradict the estimates from above that would be obtained if the behaviour of entropy numbers under real interpolation was as good as conjectured. The paper also provides sharp two-sided estimates of the entropy number e n (T) of diagonal operators T:lp® lq, T(( ak)k=1¥) = (( lkak) k=1¥) , where 0 <p <q ≤ ∞ and {li}i=1¥ is a non-increasing sequence of non-negative numbers with λ i = λ n for all i ≤ n.
Translated title of the contributionEntropy numbers and interpolation
Original languageEnglish
Pages (from-to)963 - 977
Number of pages15
JournalMathematische Annalen
Volume351
Issue number4
DOIs
Publication statusPublished - Dec 2011

Bibliographical note

Publisher: Springer

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