
Author(s) 
Huijsmans, C.B. (ed.) Luxemburg, W.A.J. (ed.) 

Title  POSITIVE OPERATORS AND SEMIGROUPS ON BANACH LATTICES 
Publisher  Kluwer Academic Publishers 
Year of publication  1992 
Reviewed by  P.P. Zabrejko 
This book contains Contributions of the Caribbean Mathematics Foundation Conference (1822 June 1990, Curacao, Netherlands Antiles) devoted to some modern aspects of the linear positive operators theory. The conference was organised by C.B. Huijsmans, W.A.J. Luxemburg with the main purpose to bring together a group of likeminded specialists from USA and Western Europe to present their recent results and to discuss their research interests. The main topics of this conference are related to the investigations of the lattice ordered Banach algebra of order bounded operators in Banach lattices, some spectral properties of linear positive operators and theory of semigroups of linear positive operators. The book contains 14 lecturearticles with small Problem Sections with a series of unresolved problems in field.
The article by Y.A. Abramovich, C.D. Aliprantis, O. Burkinshaw "Positive Operators on Krein Spaces" is a small survey of the remarkable spectral properties of positive linear operators, acting on a Banach space ordered with a solid cone; although this survey is of interest to specialists in field many important ideas and conceptions of the M. Krein  M. Rutman  M. Krasnosel'sky theory of linear positive operator in Banach spaces with cone are either distorted or omitted and so does not allow to get real representation about the uptodate state of the field. The article by Y.A. Abramovich, W. Filter "A Remark on the Representation of Vector Lattices as Spaces of Continuous RealValued Functions" presents an elegant complete characterization of Archimedean vector lattices allowing the representations as vector sublattices of bounded functions in the universally complete vector lattice of all extended realvalued continuous functions on an exremally disconnected compact Hauedorff space. The article W. Arendt, J. Voigt "Domination of Uniformly Continuous Semigroups" deals with the following result concerning semigroups in a real or complex Banach lattice E: if for some bounded operator B and C0semigroup T(t) the inequality <formula> holds then B is a regular operator and the generator A of T is bounded. The article by S.J. Bernau "Sums and Extensions of Vector Lattice Homomorphisms" is a summary account of results which characterize order bounded linear operators which are sums of lattice homomorphisms or orthomorphisms and of theorems concerning extensions of vector lattice homomorphisms. The article by B. Eberhardt, G. Greiner "Baillon's Theorem on Maximal Regularity" presents a simple proof of important Baillon's theorem on the differentiability on (0, °) of the Cauchy operator for C0semigroup in a Banach space. The article by A.W. Hager, J. Martinez "FractionDense Algebras and Spaces" is devoted to some properties of spaces X for which C(X) is fractiondense or in other words the space of minimal prime ideals in C(X) is compact and extremely disconnected. The article by C.B. Huijsmans, W.A.J. Luxemburg "An Alternative Proof of a RadonNikodym Theorem for Lattice Homomorphisms" presents a new proof of the LuxemburgSchep theorem about the equivalence of properties <formula> and <formula> for <formula> and <formula> where E is an Archimedean and F a Dedekind complete vector lattices. The article by C.B. Huijsmans, B. de Pagter "Some Remarks on Disjointness Preserving Operators" presents the simple proof of the following result: if T: E > E is a lattice homomorphism on a Banach lattice E, then i) sigma(T) = {1} implies T = I; and ii) r(T  I) < 1 implies T elem Z(E), the center of E. The article by L. Maligranda "Weakly Compact Operators and Interpolation" is a interesting survey of some uptodate results about weakly compact operators and their interpolations. The article by P. MeyerNieberg "Aspects of Local Spectral Theory for Positive Operators" deals with the positive solvability of the equation <formula> in fa Banach space E in the following cases: i) <formula>, ii). the norm in E is replaced by a nonequivalent one; iii), the domain T is a dense ideal in E. The article by B. de Pagter "A WienerYoung Type Theorem for Dual Semigroups" presents some generalization of the WienerYoung theorem on the equality <formula> (mu is a complex bounded Borel measure, mut is its tshift, mus is its singular part) for strongly continuous semigroup of positive operators in Banach lattices. The article by A.R. Schep "Krivine's Theorem and Indices of a Banach Lattice" presents an exposition of all details of a proof of Krivine's theorem for the upper and lower indices of a Banach lattices which describe some subspaces of any finite dimension in these lattices that are isomorphic to spaces <formula>. The article by A.W. Wickstead "Representations of Archimedean Riesz Spaces by Continuous Functions" is a brief survey of representations of Archimedean Riesz spaces in spaces of continuous extended realvalued functions with some applications in the general theory of the Riesz spaces. The article by X.D. Zhang "Some Aspect of the Aspects of the Spectral Theory of Positive Operators" deals with the following problem about additional conditions under that the equality sigma(T) = {1} for a positive operator implies the inequality T >= I.
The small Problem Section contains 16 actual problems concerning some questions that are discussed in this book.
In general the book is of interest to specialists whose work involves the theory of ordered Banach spaces, and in particular, Banach function spaces, theory of linear positive operators and applications of these theories to oneparameter semigroups and partial differential equations, probability theory, control theory and so on.