On some vector lattices of operators and their finite elements

Chapter 2 Positive operators

It is known that if a rearrangement invariant function space E on (0,1) has an unconditional basis then each linear continuous operator on E is a sum of two narrow operators. On the other hand, the sum of two narrow operators in L1 is narrow. To find a general approach to these results, we extend the notion of a narrow operator to the case when the domain space is a vector lattice. Our main result asserts that the set Nr(E, F )o f all nar- row regular operators is a band in the vector lattice Lr(E, F) of all regular operators from a non-atomic order continuous Banach lattice E to an order continuous Banach lattice F. The band generated by the disjointness preserv- ing operators is the orthogonal complement to Nr(E, F )i nLr(E, F). As a consequence we obtain the following generalization of the Kalton-Rosenthal theorem: every regular operator T : E → F from a non-atomic Banach lattice E to an order continuous Banach lattice F has a unique representation as T = TD + TN where TD is a sum of an order absolutely summable family of disjointness preserving operators and TN is narrow. Mathematics Subject Classification (2000). Primary 47B65; secondary 47B38.

A general extended functional is a functional which is allowed to take on infinite values; in other words such functionals are similar to the Lebesgue integral on the space of all integrable functions. The problem of representing a general extended functional f on a vector lattice X as an operator with finite values only has been solved in [12]. In fact, this problem has been solved for a larger class of general extended operators on an ordered vector space. The solution of this problem was given by means of an extension of the range R of the functional f to some ultrapower of R. Notice, however, that it is not always the case that a functional f can be considered as a trace of some internal functional *f:*X→R. (We remark, without going into details, that such an internal functional exists exactly in the case when the results of the nonstandard analysis can be used for investigation of the given functional.) In [12] a ‘standard’ necessary and sufficient condition was given for solving this latter problem on the existence of *f. Namely, f is a trace of an internal *f if and only if it is finitely extendable. This result makes the finite extendability problem worthy of study. The present paper is a first attempt in this direction. Simultaneously we introduce and study a more general notion of weak finite extendability that coincides with finite extendability, for instance, for vector lattices with a strong unit.

An operator T from a Banach space E into a Banach lattice F is said to be semi-compact if for each e > 0, there exists some u ∈ F+ such that T (BE) ⊂ [−u, u] + eBF where BH is the closed unit ball of H = E or F and F+ = {x ∈ F : 0 ≤ x}. Contrarily to compact operators, the class of semi-compact operators satisfies the domination problem. Indeed, if S and T are operators from a Banach lattice E into F such that 0 ≤ S ≤ T and T is semi-compact, then S is semi-compact (Theorem 18.20 of [3]). It is clear that each compact operator is semi-compact but a semicompact operator is not necessary compact. In fact, the operator identity Idl∞ : l∞ −→ l∞ is semi-compact but it is not compact. Let us recall that Dodds and Fremlin ([11], Theorem 125.5) proved that if E and F are two Banach lattices such that the norms of F and of the topological dual E ′ are order continuous, then each semi-compact operator T from E into F which is AM-compact (i.e. a regular operator such that the image of each order bounded subset of E is a relatively compact subset of F ) is compact. In [5], we studied the compactness of positive Dunford-Pettis operators, and we gave several characterizations of Banach lattices on which each

Let A" be a set, A a family of real-valued functions on X which contains the constants, pA the weak uniformity generated by A, and U{nAX) the collection of uniformly continuous functions to the real line R.The problem is how to construct U(\iAX) from A. The main result here is: For A a vector lattice, the collection of suprema of countable, finitely Aequiuniform, order-one subsets of A + is uniformly dense in U(pAX).Two less technical corollaries: If A is a vector lattice (resp., vector space), then the collection of functions which are finitely A-uniform and uniformly localfy-A (resp., uniformly locally piecewise-A) is uniformly dense in U(pAX).Further, for any A, a finitely A-uniform function is fust a composition F (a,,... ,af) for some a{.^ e A and F uniformly continuous on the range of (fli.ap) in Rf.Thus, such compositions are dense in U(iiAX).For BU(iiAX), the compositions with F e BU(Rr) are dense (B denoting bounded functions).So, in a sense, to know U(pAX) it suffices to know A and subspaces of the spaces Rp, and to know BU(pAX), A and the spaces Rr suffice.In case A is a vector lattice and A ** BA (i.e., A consists of bounded functions), the problem of describing U(pAX) in terms of A has an elegant solution: A is uniformly dense in U(pAX) (i.e., if / G U(pAX) and t > 0 there is a E A with \f(x) -a(x)\ < e for all x E A").This can be seen to be essentially equivalent to the usual Stone-Weierstrass Theorem.The result appears to have been first published in 1955 by Maak [M] and Nbeling and Bauer [NB]; we give a short proof in the course of proving our main theorem.If A contains unbounded functions, the situation is more complicated: A = U(R)\N is a "uniformly closed" vector lattice, while pA is (uniformly) discrete.With X = R, let A be the piecewise linear functions (finitely many pieces); pA is the usual uniformity on R and the closure of A consists only of continuous eventually linear functions.

In this paper, we continue investigation of the lateral order on vector lattices started in [25]. We consider the complexification of a real vector lattice E and introduce the lateral order on . Our first main result asserts that the set of all fragments of an element of the complexification of an uniformly complete vector lattice E is a Boolean algebra. Then, we study narrow operators defined on the complexification of a vector lattice E, extending the results of articles [22, 27, 28] to the setting of operators defined on complex vector lattices. We prove that every order‐to‐norm continuous linear operator from the complexification of an atomless Dedekind complete vector lattice E to a finite‐dimensional Banach space X is strictly narrow. Then, we prove that every C‐compact order‐to‐norm continuous linear operator from to a Banach space X is narrow. We also show that every regular order‐no‐norm continuous linear operator from to a complex Banach lattice is narrow. Finally, in the last part of the paper we investigate narrow operators taking values in symmetric ideals of compact operators.

Unbounded order convergence in dual spaces

We investigate properties of order continuous operators on pre-Riesz spaces with respect to the embedding of the range space into a vector lattice cover or, in particular, into its Dedekind completion. We show that order continuity is preserved under this embedding for positive operators, but not in general. For the vector lattice \ell_0^\infty of eventually constant sequences, we consider the pre-Riesz space of regular operators on \ell_0^\infty and show that making the range space Dedekind complete does not provide a vector lattice cover of the pre-Riesz space. A similar counterexample is obtained for the directed part of the space of order continuous operators on \ell_0^\infty .

Riesz homomorphisms between vector lattices are characterized by interval preserving order adjoints. Buskes, van Rooij, and van Haandel generalized the notion of a Riesz homomorphism to linear maps between general ordered vector spaces. In this paper, we discuss how far the characterization between vector lattices extends to directed ordered vector spaces and pre-Riesz spaces.

R. Drnovšek, D. Kokol-Bukovšek, L. Livshits, G. MacDonald, M. Omladič, and H. Radjavi constructed an irreducible set of positive nilpotent operators on $L^p[0,1)$ which is closed under multiplication, addition and multiplication by positive real scalars with the property that any finite subset is ideal-triangularizable. In this paper we prove the following: every algebra of nilpotent operators which is generated by a set of positive operators on a Banach lattice is ideal-triangularizable whenever the nilpotency index of its operators is bounded; every finite subset of an algebra of nilpotent operators which is generated by a set of positive operators on a Banach lattice is ideal-triangularizable.

On the ring ideal generated by a positive operator

The main aim of this paper is studying the family $$W_b(E,F)$$ of b-weakly compact operators between two Banach lattices. For an order dense sublattice G of a vector lattice E, if $$T:G\rightarrow F$$ is a b-weakly compact operator between two Banach lattices, then $$T\in W_b(E,F)$$ whenever the norm of E is order continuous and $$T:E\rightarrow F$$ is a positive operator. We also investigate the relationship between $$W_b(E,F)$$ and some other classes of operators like $$L^{(1)}_c(E,F)$$ and $$L^{(2)}_c(E,F)$$ .

Throughout this paper we denote by L p the Banach lattice of p-integrable functions on a σ-finite measure space (X, B, μ), where 1 ≤ p ≤ ∞. We will consider those aspects of the theory of positive linear operators, which are in some way special due to the fact the operators are acting on L p-spaces. For general information about positive operators on Banach lattices we refer to the texts [1]. [20], and [36]. Our focus on L p-spaces does not mean that in special cases some of the results can not be extended to a larger class of Banach lattices of measurable function such as Orlicz spaces or re-arrangement invariant Banach function spaces. However in many cases the results in these extensions are not as precise or as complete as in the case of L p-spaces. We will discuss results related to the boundedness of positive linear operators on L p-spaces. The most important result is the so-called Schur criterion for boundedness. This criterion is the most frequently used tool to show that a concrete positive linear operator is bounded from L p to L q. Then we will show how this result relates to the change of density result of Weis [33]. Next the equality case of Schur’s criterion is shown to be closely related to the question whether a given positive linear operator attains its norm. We discuss in detail the properties of norm attaining operators on L p-spaces and discuss as an example the weighted composition operators on L p-spaces. Then we return to the Schur criterion and show how it can be applied to the factorization theorems of Maurey and Nikisin. Most results mentioned in this paper have appeared before in print, but sometimes only implicitly and scattered over several papers. Also a number of the proofs presented here are new.

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.

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.