On the structure of semigroups on \(L_{p}\) with a bounded \(H^{\infty}\)-calculus
type: Item
instance of: scholarly article
zbMATH Open document ID: 1311.47017
author: Stephan Fackler
published in: Bulletin of the London Mathematical Society
publication date: 2014-10-15T00:00:00Z
full work available at URL: https://arxiv.org/abs/1310.4672
Mathematics Subject Classification ID: 47A60
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 46L07
Mathematics Subject Classification ID: 47L25
zbMATH DE Number: 6355604
zbMATH Keywords: bounded analytic semigroups
zbMATH Keywords: \(L_p\)-spaces
zbMATH Keywords: (UMD)-Banach lattices
MaRDI profile type: Publication
OpenAlex ID: W3102200281
arXiv ID: 1310.4672
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DOI: 10.1112/BLMS/BDU062
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title: On the structure of semigroups on \(L_{p}\) with a bounded \(H^{\infty}\)-calculus
review text: The author considers bounded analytic semigroups on a reflexive \(L_p\)-space and shows the following converse of Weis' result, generalizing Le Merdy's theorem from the Hilbert space case to an \(L_p\)-setting: for a semigroup \((T (t ))_{t \geq 0}\) on \(L_p\) \(( p \in (1, \infty))\) whose negative generator has a bounded \(H^\infty (\Sigma_{\theta})\)-calculus for some \(\theta < {\pi}/{2}\) there exists a bounded analytic semigroup \(( R( z ))\) on some bigger \(L_p\)-space, which is positive and contractive on the real line, \(( R( z ))\)-invariant closed subspaces \(N \subset M\) and an isomorphism \(S\) from \(L_p\) onto \(M / N\) such that \(T (t )= S^{-1}\hat R(t )S\) for all \(t \geq 0\), where \((\hat R(t ))_{t \geq 0}\) is the induced semigroup on \(M / N\).NEWLINENEWLINEAn analogous result is also obtained for semigroups on subspace quotients of general unconditional martingale difference (UMD)-Banach lattices.NEWLINENEWLINESeveral useful remarks illustrating the problem on general (UMD)-Banach spaces and lattices are included. Moreover, some open problems associated with the subject are proposed.
reviewed by: Miklavž Mastinšek
description: scientific article; zbMATH DE number 6355604