The adjoint of a semigroup of linear operators
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 \(X^ \odot\) has the Radon-Nikodym property (RNP). Two typical results are: (1) If \(X'\) does not have the RNP then at least one of the quotient \(X^{\odot\odot}/X\) and \(X'/X^{\odot\odot}\) is non-separable. (2) If \(\mathbb{T}\) is \(\odot\)-reflexive and if the Favard class \(\text{Fav}(\mathbb{T})\) equals \(\mathbb{D}(A)\) then \(X\) has the RNP. It is well-known that translation of a bounded measure \(\mu\) on \(\mathbb{R}\) (i.e. the function \(t\to\mu_ t\) where \(\mu_ t(E)=\mu(E+t)\)) is continuous iff \(\mu\) is absolutely continuous with respect to Lebesgue measure. In the senventh chapter the author generalizes this result to vector measures using the sun-dual of the tensor product \(\mathbb{T}\otimes Id\) on the Banach space \(X\otimes Y\) where in particular the projective, the injective and the \(I\)-tensor product are considered. A typical result is the following one: Assume that \(R(\lambda,A)\) is compact. Then the sun-dual of \((T_ t\otimes I)\) on \(X\otimes_ \pi Y\) is \((X\otimes_ \pi Y)\odot= X^ \odot\otimes_ \varepsilon Y'\). Finally the last chapter is devoted to the sun-dual of semigroups of positive operators on a Banach lattice \(E\). Only little can be said in this case, since the sun-dual need not be a Banach lattice. But if the dual lattice \(E'\) has order continuous norm then \(E^ \odot\) is a projection band of \(E'\). As a more special result a vector valued version of a theorem of Wiener and Young concerning the translates of singular measures on \(\mathbb{R}\) is obtained. Only very few misprints are to be found which may irritate the reader, e.g. 3rd line from above on p. 56 where one of the \(x^ \odot\) has to be cancelled, and condition (iii) of the Theorem 3.2.8 on p. 57 where the fixed index \(n\) and the index running to \(\infty\) are the same. One may miss some concrete applications which the author refused to incorporate into this volume. I got knowledge of a very nice one quite recently [\textit{R. Hempel} and \textit{J. Voigt}: ``The spectrum of Schrödinger operators in \(L_ p(\mathbb{R}^ d)\) and in \(C_ 0(\mathbb{R})^ d)\)'', preprint (1993)]. But a lot of others will follow if they are not already implicitely included in the book. The book is written up very clearly and presents many results from different people including those of the author in an elegant manner. So it will serve as a concise reference for the included material as well as a starting point for further research.
perturbation of infinitesimal generators
strongly continuous semigroup of bounded linear operator
theorem of Hille and Phillips and de Pagter
sun-dual semigroups of positive operators on a Banach lattice
vector valued version of ta theorem of Wiener and Young concerning the translates of singular measures
scientific article; zbMATH DE number 108530