Banach Lattices and Positive Operators

TL;DR: In this paper, the authors propose the use of linear operators on positive matrices and apply it to non-positive matrices, including the case of positive projections. But they do not consider the case where positive projections are defined by a linear operator.

Abstract: 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.