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.