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Book review

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 (18-22 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 lecture-articles 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 up-to-date state of the field. The article by Y.A. Abramovich, W. Filter "A Remark on the Representation of Vector Lattices as Spaces of Continuous Real-Valued 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 real-valued 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 C0-semigroup 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 C0-semigroup in a Banach space. The article by A.W. Hager, J. Martinez "Fraction-Dense Algebras and Spaces" is devoted to some properties of spaces X for which C(X) is fraction-dense 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 Radon-Nikodym Theorem for Lattice Homomorphisms" presents a new proof of the Luxemburg-Schep 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 up-to-date results about weakly compact operators and their interpolations. The article by P. Meyer-Nieberg "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 non-equivalent one; iii), the domain T is a dense ideal in E. The article by B. de Pagter "A Wiener-Young Type Theorem for Dual Semigroups" presents some generalization of the Wiener-Young theorem on the equality <formula> (mu is a complex bounded Borel measure, mut is its t-shift, 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 real-valued 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 one-parameter semigroups and partial differential equations, probability theory, control theory and so on.