Maps with separable dynamics and the spectral properties of the operators generated by them. (31st March 2015)

• Record Type: Journal Article
• Authors: Antonevich, A B; Ahmatova, A A; Makowska, J
• Is Part Of: Sbornik. Volume 206:Number 3(2015)
• Page Start: 341
• Page End: 369
• Publication Date: 2015-03-31
• Subjects: Mathematics -- Periodicals; 510.5
• DOI: 10.1070/SM2015v206n03ABEH004461
• Languages: English
• ISSNs: 1064-5616

Abstract

<abstract xml:lang="en"> <title>Abstract</title> <p>A map <inline-graphic xlink:href="ark:/27927/pgj11880d6r" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink"><alt-text>$ \alpha $</alt-text></inline-graphic> of a space <inline-graphic xlink:href="ark:/27927/pgj11880d79" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink"><alt-text>$ X$</alt-text></inline-graphic> into itself generates weighted shift operators <inline-graphic xlink:href="ark:/27927/pgj11880d56" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink"><alt-text>$ B$</alt-text></inline-graphic> in function spaces on <inline-graphic xlink:href="ark:/27927/pgj11880d79" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink"><alt-text>$ X$</alt-text></inline-graphic>. The spectral properties of such operators are intimately connected with the dynamics of <inline-graphic xlink:href="ark:/27927/pgj11880d6r" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink"><alt-text>$ \alpha $</alt-text></inline-graphic>. It was known previously that the spectrum of an operator depends only on the set of invariant ergodic measures for <inline-graphic xlink:href="ark:/27927/pgj11880d6r" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink"><alt-text>$ \alpha $</alt-text></inline-graphic>. Conditions for the right invertibility of the operators <inline-graphic xlink:href="ark:/27927/pgj11880d33" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink"><alt-text>$ B-\lambda I$</alt-text></inline-graphic> are obtained when <inline-graphic xlink:href="ark:/27927/pgj11880d4n" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink"><alt-text>$ \lambda$</alt-text></inline-graphic> is a spectral value. The main result states that right invertibility is only possible when a nontrivial attractor exists.</p> <p>Bibliography: 29 titles. </p> </abstract>