Introduction to Operator Theory in Riesz Spaces

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.