On lattice properties of the composition operator

Lebesgue-type convergence theorems in Banach lattices with applications to compact operators

Convergence of Riesz space martingales

Let Z be a Banach lattice endowed with positive cone C and an order-continuous norm j.j .Let G be a left-amenable semigroup of positive linear endomorphisms of Z .Then the positive fixed points Co of Z under G form a lattice cone, and their linear span Z0 is a Banach lattice under an order-continuous norm ||.||0 which agrees with |.| on Co.A counterexample shows that under the given conditions Z0 need not contain all the fixed points of Z under G , and need not be a sublattice of (Z, C).The paper concludes with a discussion of some related results.Let G be a semigroup.We denote by m(C7) the Banach space of all bounded linear functions from G into the real numbers R, under the supremum norm.We denote by m'iG) the Banach dual of w(C7).With each T g G we associate an endomorphism Tm of /n(C7) defined by iTmb)iU) = biTU) for U G G and b G m(G) where TU denotes the composite of T and U under the semigroup operation.An element p G m*iG) is called a mean for G iff inf biT) < pib) < sup biT) for all b G G TG TG and left-invariant for G iff T'mp=p forallTGG, where T'm denotes the adjoint of Tm .Following M. Day [1, p. 108] we call the semigroup G left-amenable iff there exists a left-invariant mean for G.In particular, any Abelian semigroup is left-amenable [1, Theorem 4, p. 108].A Banach lattice is said to have order-continuous norm iff every decreasing sequence of positive elements is norm convergent [3, 5.12, p. 92; 5.10(d), p. 89].Theorem.Let (Z,C, ||.||) be a Banach lattice with order-continuous norm.Let G be a left-amenable semigroup of positive linear operators from Z into Z.Define C0 = {x G C : Tx = x for all T gG} , Z0 = C0 -CQ.

In this paper we formulate and prove analogues of the Hahn-Jordan decomposition and an Ando-Douglas-Radon-Nikodým theorem in Dedekind complete Riesz spaces with a weak order unit, in the presence of a Riesz space conditional expectation operator. As a consequence we can characterize those subspaces of the Riesz space which are ranges of conditional expectation operators commuting with the given conditional expectation operators and which have a larger range space. This provides the first step towards a formulation of Markov processes on Riesz spaces.

The sup-completion of a Dedekind complete vector lattice

Unbounded order convergence in dual spaces

For Banach lattices E and F, let P, Q : E → F be n-homogeneous polynomials such that 0 ≤ Q ≤ P. We prove that if F has an order continuous norm and P is weakly compact then Q is also weakly compact. We also prove that if F is an atomic Banach lattice with an order continuous norm and P is compact then Q is also compact.

In terms of quasi-interior points, criteria that a Banach lattice has order continuous norm or is an -space with a unit are given. For example, if is Dedekind complete and has a weak order unit, then has order continuous norm if and only if the set of quasi-interior points of coincides with the set of weak order units of ; a Banach lattice is an -space with a unit if and only if the set of all quasi-interior points of coincides with the set { : for some > 0}. Analogous questions are considered for the case of an ordered Banach space with a cone . Moreover, it is shown that every nonzero point of a cone = {0} is quasi-interior if and only if dim = 1. We also study various D-properties of a cone ; in particular, the conditions for which the relation = 0 with > 0 implies that is not a quasi-interior point are considered.

In Definition 3 of this article, a condition has been omitted. Definition 3 should read (i.e. the italicized words below were omitted): Definition 3 Let E be a Dedekind complete Riesz space with conditional expectation T and weak order unit e = T e. Let P and Q be band projections on E , we say that P and Q are T -conditionally independent with respect to T if TPTQe = TPQe = TQTPe. (1) We say that twoRiesz subspaces E1 and E2 of E , containingR(T ), areT -conditionally independent with respect to T if all band projections Pi , i = 1, 2, in E with Pi e ∈ Ei , i = 1, 2, are T -conditionally independent with respect to T .

In this paper we study the structure of the set {\rm Hom}(X,\mathbb{R}) of all lattice homomorphisms from a Banach lattice X into \mathbb{R} . Using the relation among lattice homomorphisms and disjoint families, we prove that the topological dual of the free Banach lattice {\rm FBL}(A) generated by a set A contains a disjoint family of cardinality 2^{|A|} , answering a question of B. de Pagter and A. W. Wickstead. We also deal with norm-attaining lattice homomorphisms. For classical Banach lattices, as c_0 , L_p - and C(K) -spaces, every lattice homomorphism on it attains its norm, which shows, in particular, that there is no James theorem for this class of functions. We prove that, indeed, every lattice homomorphism on X and C(K,X) attains its norm whenever X has order continuous norm. On the other hand, we provide what seems to be the first example in the literature of a lattice homomorphism which does not attain its norm. In general, we study the existence and characterization of lattice homomorphisms not attaining their norm in free Banach lattices. As a consequence, it is shown that no Bishop–Phelps type theorem holds true in the Banach lattice setting, i.e., not every lattice homomorphism can be approximated by norm-attaining lattice homomorphisms.

We give conditions for the linear span of the positive L-weakly compact (resp. M-weakly compact) operators to be a Banach lattice under the regular norm, for that Banach lattice to have an order continuous norm, to be an AL-space or an AM-space.

On the ring ideal generated by a positive operator

We give a characterization of those cr-Dedekind complete Banach lattices for which every continuous linear operator T: E -> co is a difference of two positive linear operators from E into cq .

Über die konvergenz positiver operatoren mit werten in atomaren Banachverbänden mit ordnungsstetiger norm

In this paper, we first give a new definition of Ω-Dedekind complete Riesz space (E,≤) in the frame of vector metric space (Ω,ρ,E) and we investigate the relation between Dedekind complete Riesz space and our new concept. Moreover, we introduce a new contraction so called α-vector proximal contraction mapping. Then, we prove certain best proximity point theorems for such mappings in vector metric spaces (Ω,ρ,E) where (E,≤) is Ω-Dedekind complete Riesz space. Thus, for the first time, we acquire best proximity point results on vector metric spaces. As a result, we generalize some fixed point results proved in both vector metric spaces and partially ordered vector metric spaces such as main results of V4 . Further, we provide nontrivial and comparative examples to show the effectiveness of our main results.