Vector lattices-also called Riesz spaces, K-lineals, or linear lattices-were first considered by F. Riesz, L. Kantorovic, and H. Freudenthal in the middle nineteen thirties; thus their early theory dates back almost as far as the beginning of the systematic investigation of Banach spaces. Schools of research on vector lattices were subsequently founded in the Soviet Union (Kantorovic, Judin, Pinsker, Vulikh) and in Japan (Nakano, Ogasawara, Yosida); other important contri- butions came from the United States (G. Birkhoff, Kakutani, M. H. Stone). L. Kantorovic and his school first recognized the importance of studying vector lattices in connection with Banach's theory of normed vector spaces; they investigated normed vector lattices as well as order-related linear operators between such vector lattices. (Cf. Kantorovic-Vulikh-Pinsker [1950] and Vulikh [1967].)
However, in the years following that early period, functional analysis and vector lattice theory began drifting more and more apart; it is my impression that "linear order theory" could not quite keep pace with the rapid development of general functional analysis and thus developed into a theory largely existing for its own sake, even though it had interesting and beautiful applications here and there.
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Für höhere Schule und Studium
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ISBN-13
978-3-540-06936-2 (9783540069362)
DOI
10.1007/978-3-642-65970-6
Schweitzer Klassifikation
I. Positive Matrices.- § 1. Linear Operators on ?n.- § 2. Positive Matrices.- § 3. Mean Ergodicity.- § 4. Stochastic Matrices.- § 5. Doubly Stochastic Matrices.- § 6. Irreducible Positive Matrices.- § 7. Primitive Matrices.- § 8. Invariant Ideals.- § 9. Markov Chains.- § 10. Bounds for Eigenvalues.- Notes.- Exercises.- II. Banach Lattices.- § 1. Vector Lattices over the Real Field.- § 2. Ideals, Bands, and Projections.- § 3. Maximal and Minimal Ideals. Vector Lattices of Finite Dimension.- § 4. Duality of Vector Lattices.- § 5. Normed Vector Lattices.- § 6. Quasi-Interior Positive Elements.- § 7. Abstract M-Spaces.- § 8. Abstract L-Spaces.- § 9. Duality of AM- and AL-Spaces. The Dunford-Pettis Property.- § 10. Weak Convergence of Measures.- § 11. Complexification.- Notes.- Exercises.- III. Ideal and Operator Theory.- § 1. The Lattice of Closed Ideals.- § 2. Prime Ideals.- § 3. Valuations.- § 4. Compact Spaces of Valuations.- § 5. Representation by Continuous Functions.- § 6. The Stone Approximation Theorem.- § 7. Mean Ergodic Semi-Groups of Operators.- § 8. Operator Invariant Ideals.- § 9. Homomorphisms of Vector Lattices.- § 10. Irreducible Groups of Positive Operators. The Halmos-von Neumann Theorem.- § 11. Positive Projections.- Notes.- Exercises.- IV. Lattices of Operators.- § 1. The Modulus of a Linear Operator.- § 2. Preliminaries on Tensor Products. New Characterization of AM- and AL-Spaces.- § 3. Cone Absolutely Summing and Majorizing Maps.- § 4. Banach Lattices of Operators.- § 5. Integral Linear Mappings.- § 6. Hilbert-Schmidt Operators and Hilbert Lattices.- § 7. Tensor Products of Banach Lattices.- § 8. Banach Lattices of Compact Maps. Examples.- § 9. Operators Defined by Measurable Kernels.- § 10. Compactness of Kernel Operators.- Notes.- Exercises.- V. Applications.- § 1. An Imbedding Procedure.- § 2. Approximation of Lattice Homomorphisms (Korovkin Theory).- § 3. Banach Lattices and Cyclic Banach Spaces.- § 4. The Peripheral Spectrum of Positive Operators.- § 5. The Peripheral point Spectrum of Irreducible Positive Operators.- § 6. Topological Nilpotency of Irreducible Positive Operators.- § 7. Application to Non-Positive Operators.- § 8. Mean Ergodicity of Order Contractive Semi-Groups. The Little Riesz Theorem.- Notes.- Exercises.- Index of Symbols.
