A characterization of sun-reflexivity
type: Item
instance of: scholarly article
title: A characterization of sun-reflexivity
zbMATH Open document ID: 0696.47039
DOI: 10.1007/BF01442743
author: B. de Pagter
published in: Mathematische Annalen
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164523
review text: The dual semigroup \(\{T^*(t)\}_{t\geq 0}\) of a strongly continuous semigroup \(\{T(t)\}_{t\geq 0}\) of bounded linear operators (i.e. \(C_ 0\)-semigroup) on a Banach space X need not be strongly continuous in \(X^*\). Let \(X^{\odot}\) denote the maximal invariant subspace for \(T^*(\cdot)\) such that the restriction \(T^{\odot}(\cdot)=T^*(\cdot)|_{X^{\odot}}\) is a \(C_ 0\)- semigroup. Let j be the canonical embedding of X into \(X^{**}\), and \(\gamma_{\odot}\) be the restriction mapping, sending each \(x^{**}\in X^{**}\) to \(x^{**}| X^{\odot}\). Then the mapping \(j_{\odot}=\gamma_{\odot}\circ j\) is a norm isomorphism form X into \(X^{\odot \odot}\). In general, it is not onto. If it is, X is called reflexive. It is known [cf. \textit{E. Hille} and \textit{R. S. Phillips}, Functional Analysis and Semigroups, Am. Math. Soc. Colloquium Pubs. vol. 31 (1957; Zbl 0078.100)] that \(\odot\)-reflexivity of X is equivalent to \(\sigma (X,X^{\odot})\)-compactness of \((\lambda -A)^{-1}\), \(\lambda\in \rho (A)\), where A denotes the infinitesimal generator of T(\(\cdot)\). The present paper shows that it is also equivalent to the weak compactness of \((\lambda -A)^{-1}\), and, whenever X has the Dunford-Pettis property (e.g. \(L^ 1\)-spaces), \(\odot\)-reflexivity of X is even equivalent to the compactness of \((\lambda -A)^{-1}\).
reviewed by: Sen-Yen Shaw
Mathematics Subject Classification ID: 47D03
zbMATH DE Number: 4140599
zbMATH Keywords: sun-reflexivity
zbMATH Keywords: dual semigroup
zbMATH Keywords: strongly continuous semigroup
zbMATH Keywords: \(C_ 0\)- semigroup
zbMATH Keywords: maximal invariant subspace
zbMATH Keywords: Dunford-Pettis property
MaRDI profile type: Publication
cites work: Dual semigroups and second order linear elliptic boundary value problems
cites work: Perturbation theory for dual semigroups. I. The sun-reflexive case
cites work: Q3906660
cites work: Linear Operations on Summable Functions
cites work: Q3254057
cites work: Sur Les Applications Lineaires Faiblement Compactes D'Espaces Du Type C(K)
cites work: Q3244654
cites work: The adjoint semi-group
cites work: Q4049667
Recommended article: Hahn-Banach type theorems for adjoint semigroups
Recommended article: Sandwiching C 0 -Semigroups
Recommended article: Continuity of the restriction of \(C_0\)-semigroups to invariant Banach subspaces
Recommended article: Relative compactness of orbits and geometry of Banach spaces
Recommended article: Extremal properties of contraction semigroups on \(c_ 0\)
Recommended article: Optimal properties of contraction semigroups in Banach spaces
Recommended article: Q5315784
Recommended article: Reflexivity, the dual Radon-Nikodym property, and continuity of adjoint semigroups
Recommended article: Viability for nonautonomous semilinear differential equations
Recommended article: Slowly Changing Vectors and the Asymptotic Finite-Dimensionality of an Operator Semigroup
description: scientific article; zbMATH DE number 4140599