Introduction to Operator Theory in Riesz Spaces

ISBN-13: 9783642644870 / Angielski / Miękka / 2011 / 312 str.

Adriaan C. Zaanen

Opis

Since the beginning of the thirties a considerable number of books on func tional analysis has been published. Among the first ones were those by M. H. Stone on Hilbert spaces and by S. Banach on linear operators, both from 1932. The amount of material in the field of functional analysis (in cluding operator theory) has grown to such an extent that it has become impossible now to include all of it in one book. This holds even more for text books. Therefore, authors of textbooks usually restrict themselves to normed spaces (or even to Hilbert space exclusively) and linear operators in these spaces. In more advanced texts Banach algebras and (or) topological vector spaces are sometimes included. It is only rarely, however, that the notion of order (partial order) is explicitly mentioned (even in more advanced exposi tions), although order structures occur in a natural manner in many examples (spaces of real continuous functions or spaces of measurable function ). This situation is somewhat surprising since there exist important and illuminating results for partially ordered vector spaces, in . particular for the case that the space is lattice ordered. Lattice ordered vector spaces are called vector lattices or Riesz spaces. The first results go back to F. Riesz (1929 and 1936), L. Kan torovitch (1935) and H. Freudenthal (1936)."

Spis treści

1 Lattices and Boolean Algebras.- 1 Partially Ordered Sets.- 2 Lattices.- 3 Boolean Algebras.- 2 Riesz Spaces.- 4 Riesz Spaces.- 5 Equalities and Inequalities.- 6 Distributive Laws, the Birkhoff Inequalities and the Riesz Decomposition Property.- 3 Ideals, Bands and Disjointness.- 7 Ideals and Bands.- 8 Disjointness.- 4 Archimedean Spaces and Convergence.- 9 Archimedean Riesz spaces.- 10 Order Convergence and Uniform Convergence.- 5 Projections and Dedekind Completeness.- 11 Projection Bands.- 12 Dedekind Completeness.- 6 Complex Riesz Spaces.- 13 Complex Riesz spaces.- 7 Normed Riesz Spaces and Banach Lattices.- 14 Normed Spaces and Banach Spaces.- 15 Normed Riesz Spaces and Banach Lattices.- 8 The Riesz-Fischer Property and Order Continuous Norms.- 16 The Riesz-Fischer Property.- 17 Order Continuous Norms.- 9 Linear Operators.- 18 Linear Operators in Normed Spaces and in Riesz Spaces.- 19 Riesz Homomorphisms and Quotient Spaces.- 10 Order Bounded Operators.- 20 Order Bounded Operators.- 11 Order Continuous Operators.- 21 Order Continuous Operators.- 22 The Band of Order Continuous Operators.- 12 Carriers of Operators.- 23 Order Denseness.- 24 The Carrier of an Operator.- 13 Order Duals and Adjoint Operators.- 25 The Order Dual of a Riesz Space.- 26 Adjoint Operators.- 14 Signed Measures and the Radon-Nikodym Theorem.- 27 The Space of Signed Measures.- 28 The Radon-Nikodym Theorem.- 15 Linear Functionals on Spaces of Measurable Functions.- 29 Linear Functionals on Spaces of Measurable Functions.- 16 Embedding into the Bidual.- 30 Annihilators and Inverse Annihilators.- 31 Embedding into the Order Bidual.- 17 Freudenthal’s Spectral Theorem.- 32 Projection Bands and Components.- 33 Freudenthal’s Spectral Theorem.- 18 Functional Calculas and Multiplication.- 34 Functional Calculus.- 35 Multiplication.- 19 Complex Operators.- 36 Complex Operators.- 37 Synnatschke’s Theorem.- 20 Results with the Hahn-Banach Theorem.- 38 The Hahn-Banach Theorem in Normed Vector Spaces.- 39 The Hahn-Banach Theorem in Normed Riesz Spaces.- 21 Spectrum, Resolvent Set and the Krein-Rutman Theorem.- 40 Spectrum and Resolvent Set.- 41 The Krein-Rutman Theorem.- 22 Spectral Theory of Positive Operators.- 42 Irreducible Operators.- 43 The Spectrum of a Compact Irreducible Operator.- 44 The Peripheral Spectrum of a Positive Operator.

Blurb

The book deals with the structure of vector lattices, i.e. Riesz spaces, and Banach lattices, as well as with operators in these spaces. The methods used are kept as simple as possible. Almost no prior knowledge of functional analysis is required. For most applications some familiarity with the oridinary Lebesgue integral is already sufficient. In this respect the book differs from other books on the subject. In most books on functional analysis (even excellent ones) Riesz spaces, Banach lattices and positive operators are mentioned only briefly, or even not at all. The present book shows how these subjects can be treated without undue extra effort. Many of the results in the book were not yet known thirty years ago; some even were not known ten years ago.