The Positive Operators On Banach Lattices

The first results of Riesz space and positive operators go back to F. Riesz (1929 and 1936). Since then positive operator theorems have always played an essential role on the subject of functional analysis and have been applied to some fields such as mathematical physics and economics. In this paper, we mainly discuss positive operators and some relevant problems. On the one hand, we investigate C0 semigroups on Banach lattice, and obtain some properties of local spectral radius, the solution of operator equation, the decomposition of lattice space and the generators of semigroup and dual semigroup. These results play a fundamental part in solving of abstract Cauchy problem . On the other hand, we treat those aspects of the properties closely related to the Hilbert lattice and Banach lattice and consequently obtain some deep results. The theorems of positive operators of Banach lattice and positive operators are an inseparable part of the general Banach space and operator theory. These new methods offer us some new approaches to the operator theory.This paper contain four chapters. Chapter 1 deals with Co semigroups on Banach lattice as well as with their properties. First we introduce the local spectral theory for Co semigroups and gate that local spectral radius is less than spectral radius . We show that invariant Banach space for all 0 ≠ λ ∈ C and the positive operator equation (λ- T(t))y = x has a solution y ∈ E+ if and only if x. The relations of generators of Co semigroup and dual Co semigroup are given , that is, if A* is norm dual of A,A#,A are generators of and respectively, then (1) A* = A#; (2) A = A* |E . If {T(t)}t≥0 is Co semigroups of positive operators on Banach lattice, E* has a decomposition then chapter 2 discusses positive operators on Hilbert lattices . We state that if T ∈ B(H) is a positive operator, then r(T] ∈ a(T) and there exists a sequence of unit positive vectors such that as . With relations of operators and matric, we will show that the inverse A-1 of positive operator A is still a positive operator if and only if that A is a generalized permutation. Furthermore this section contains characterization of numerical rang and numerical radius. We show that there is a one-to-one correspondence between the complex symmetric compact set and the spectrum of the real operator. In the end we gather some results of ideals, bands, ideal irreducible and band irreducible. As an application we discuss the equivalent relations of the irreducible positive operators.Chapter 3 discusses the spectrum properties of positive operators on Banach lattices. We introduce some results as follows: the spectrum of a positive operator is a symmetrical set in the complex plane about the real axis; if T is a positive contraction operator on a Banach lattice such that a(T) = 1. then T = 7; if T is a positive operator on a finite dimensional Banach lattice such that a(T) = 1, then T ≥ 7; if T is compact operator and 0 ≤ 5 ≤ T, then a0(S) = a(S); if E is order complete Banach lattice, then roe(T)∈ aoe(T).Chapter 4 is mainly about the properties of regular operators. We give two examples to show that regular norm and operator norm may not equal to each other and the module for operator on Banach lattice may not exist firstly. Then the relation of regular operators , bounded operators and linear operators on Banach lattices are given, that is Lr(E,F) Lb(E,F) L(E,F); order dual, operator dual and algebra dual are related, i.e. E' C E* C E#. In the end, by using the natural embedding J, we obtain that J is lattice homomorphism and ||JT||r = ||T'||r for a regular operator T.