Banach Lattices

Frontmatter

1. Riesz Spaces

Abstract

In this section we will investigate Riesz spaces and normed Riesz spaces, and deduce some of their basic properties. Most of the results we will present are well-known and go back to the origin of the theory of Riesz spaces. Many of them are due to G. Birkhoff, H. Freudenthal, L.V. Kantorovič, and F. Riesz. We will treat these basic results not very extensively as the intention is to give a self contained introductory part which is as short as possible. The interested reader can find more about this subject in the book of Luxemburg and Zaanen (1971).

Peter Meyer-Nieberg

2. Classical Banach Lattices

Abstract

This section is concerned with C(K)-spaces and M-spaces. In particular, will we deduce those properties of C(K)-spaces which are closely related to the theory of Riesz spaces. An important result presented here is Kakutani’s representation theorem for an M-spaces with a unit. It plays still an important role in the theory of Riesz spaces, although many proofs which were originally based on this representation theorem now can be replaced by shorter and simpler lattice theoretical arguments.

Peter Meyer-Nieberg

3. Operators on Riesz Spaces and Banach Lattices

Abstract

In this section we will treat special classes of regular operators. First we will consider disjointness preserving operators which are closely related to lattice homomorphisms. We will show that every continuous disjointness preserving operator T is regular and has an absolute value such that |Tx| = |Tx| for all x ∈E ₊ . Furthermore, we will introduce the center Z(E) of E. We will apply this concept to deduce an approximation theorem for operators in the spirit of Freudenthal’s spectral theorem.

Peter Meyer-Nieberg

4. Spectral Theory of Positive Operators

Abstract

This section is concerned with spectral properties of positive linear operators on a Banach lattice. We start with classical theorems concerning the spectral behaviour of positive linear operators including the Krein-Rutman theorems. We will formulate and prove these theorems only for Banach lattices, although they hold with similar proofs for more general classes of ordered Banach spaces. The interested reader will find these general versions in Schaefer (1971), appendix.

Peter Meyer-Nieberg

5. Structures in Banach Lattices

Abstract

In this section we mainly are interested in showing characterizations of properties of subspaces of Banach lattices. Moreover we will use the theory of order weakly compact operators to prove some results for arbitrary Banach spaces. First we will recall some basic facts concerning Schauder bases and topological embeddings of c ₀.

Peter Meyer-Nieberg

Backmatter