The adjoint of a semigroup of linear operators
type: Item
instance of: scholarly article
title: The adjoint of a semigroup of linear operators
zbMATH Open document ID: 0780.47026
author: Jan van Neerven
published in: Lecture Notes in Mathematics
publication date: 1993-01-23T00:00:00Z
review text: Let \(\mathbb{T}=(T_ t)_{t\geq 0}\) be a strongly continuous semigroup of bounded linear operators on the Banach space \(X\). Then the so-called sun dual \(X^ \odot\) of \((X,\mathbb{T})\) is the largest subspace of the dual space \(X'\) on which the dual semigroup \((T_ t')\) is strongly continuous. If \(A\) is the generator of \(\mathbb{T}\), and if \(A'\) is the dual operator then \(X^ \odot= \overline{\mathbb{D}(A')}\), and \(A'\) is the generator of \(\mathbb{T}'|_{X^ \odot}\). If \(X\) is reflexive then \(X^ \odot=X'\) so that the theory of the sun dual to which this excellent monography is devoted, makes only sense in the case of nonreflexivity. That this theory is far from being trivial or unimportant is proved by the book under review. Its first chapter contains the basic facts about the sun-dual and the spectrum of the adjoint semigroup on this space. In the second chapter the author looks for properties of the topology \(\sigma(X,X^ \odot)\), constructs the sun-bidual and proves the deep theorem of Hille and Phillips and de Pagter that \((X,\mathbb{T})\) is sun-reflexive iff the resolvent \(R(\lambda,A)\) is weakly compact. The third chapter is devoted to interpolation, extrapolation, and the Favard class \(\text{Fav}(\mathbb{T})\) of \(\mathbb{T}\). In particular a necessary and sufficient condition for \(\text{Fav}(\mathbb{T})=\mathbb{D}(A)\) is obtained. Based on the results of the previous chapters a very elegant and powerful perturbation theory is developed in the fourth chapter including applications to nonlinear (semi-linear) Cauchy problems. The fifth chapter treats the problem how ``far is the dual away from the sun dual''. More precisely the quotient space \(X'/X^ \odot\) is studied. Among others the following result is obtained: \(X'/X\) is either \(\{0\}\) or non-separable. The remaining three chapters are devoted to more special structural problems: In the 6th chapter the case is considered that one of the dual spaces \(X'\) or \(X^ \odot\) h…
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 47B65
Mathematics Subject Classification ID: 47-02
zbMATH DE Number: 108530
zbMATH Keywords: perturbation of infinitesimal generators
zbMATH Keywords: strongly continuous semigroup of bounded linear operator
zbMATH Keywords: sun dual
zbMATH Keywords: dual semigroup
zbMATH Keywords: nonreflexivity
zbMATH Keywords: spectrum of the adjoint semigroup
zbMATH Keywords: sun-bidual
zbMATH Keywords: theorem of Hille and Phillips and de Pagter
zbMATH Keywords: interpolation
zbMATH Keywords: extrapolation
zbMATH Keywords: Favard class
zbMATH Keywords: nonlinear (semi-linear) Cauchy problems
zbMATH Keywords: Radon-Nikodym property
zbMATH Keywords: vector measures
zbMATH Keywords: sun-dual of the tensor product
zbMATH Keywords: sun-dual semigroups of positive operators on a Banach lattice
zbMATH Keywords: dual lattice
zbMATH Keywords: order continuous norm
zbMATH Keywords: vector valued version of ta theorem of Wiener and Young concerning the translates of singular measures
reviewed by: Manfred P. H. Wolff
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 108530