Discrete time analytic semigroups and the geometry of Banach spaces
type: Item
instance of: scholarly article
title: Discrete time analytic semigroups and the geometry of Banach spaces
zbMATH Open document ID: 1046.47040
author: Pierre Portal
published in: Semigroup Forum
publication date: 2003-08-31T00:00:00Z
review text: The present paper deals with the Cauchy problem of the form \[ \begin{cases} u_{n+1} -Tu_ n=f_ n, \\ u_ 0=0, \end{cases} \] where \(\{u_ n\}_{n=0}^{\infty}\) are unknown elements of a Banach space \(X,\) \(T\) is a bounded linear operator on \(X\) and the elements \(\{f_ n\}_{n=0}^{\infty}\) are fixed. If \(Y(X)\) is an \(X\)-valued sequence space, the problem is said to have \(Y\)-discrete maximal regularity if \((u_{n+1}-u_n)_{n\geq 0}\) belongs to \(Y(X)\) whenever \(\{f_n\}_{n\geq 0}\) belongs to \(Y(X).\) The author studies \(\ell^2\)-discrete maximal regularity and relations with other properties of the semigroup \((T^n)_{n\geq 0}.\) Also, it is shown that the existence of a copy of \(c_0\) in \(X\) is related to the existence of particular semigroups with \(\ell^\infty\)-discrete maximal regularity.
reviewed by: Ryszard Szwarc
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 46B03
Mathematics Subject Classification ID: 46C15
zbMATH DE Number: 1972516
zbMATH Keywords: semigroups of operators
zbMATH Keywords: regularity of solutions
zbMATH Keywords: discrete time semigroups
MaRDI profile type: Publication
full work available at URL: s00233-002-0009-1
OpenAlex ID: W2065954466
DOI: 10.1007/S00233-002-0009-1
description: scientific article; zbMATH DE number 1972516