Submultiplicative operators on Ck-spaces

Dmitry Faifman*, Hermann König, Vitali Milman

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

Abstract

Let I⊂ ℝ be open, let k∈ N0 and T : Ck(I) →Ck(I) be bijective, pointwise continuous and submultiplicative, i.e., (Formula presented) Suppose also that T (-1) < 0 and that T and T-1 are positivity preserving. We show that there is a homeomorphism u : I → I and that there are continuous functions p,A ∈ C(I) with p > 0, A ≥ 1 such that (Formula presented). For k ∈ N, we have A = p = 1, and u is a Ck-diffeomorphismus so that Tf(u(x)) = f(x), i.e., T is multiplicative for k ∈ N. This rigidity property also holds for supermultiplicative operators.

Original languageEnglish
Title of host publicationOperator Theory
Subtitle of host publicationAdvances and Applications
PublisherSpringer International Publishing
Pages267-279
Number of pages13
DOIs
StatePublished - 2018

Publication series

NameOperator Theory: Advances and Applications
Volume261
ISSN (Print)0255-0156
ISSN (Electronic)2296-4878

Funding

FundersFunder number
Bloom's Syndrome Foundation0361-4561
Alexander von Humboldt-Stiftung
Iowa Science Foundation826/13
Natural Sciences and Engineering Research Council of Canada500549
Minerva Foundation

    Keywords

    • Approximate indicator
    • Positivity preserving diffeomorphism
    • Submultiplicative operators

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