The classification of injective Banach lattices

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.

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

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

It is well known that a differentiable real valued function on the real line is convex iff its derivative is nondecreasing. This characterization of differentiable convex functions does not extend if the domain of the function is a Banach lattice of $\dim \geqslant 2$. In this paper a complete characterization of a subclass of differentiable convex functions on a given Banach lattice is obtained in terms of the monotonicity and natural orthogonality properties of the gradients. It is well known that a differentiable real valued function on the real line is convex iff its derivative is nondecreasing. It is also known that a ${C^1}$-real valued function $F$ on an arbitrary Banach space $E$ is convex iff the gradient of $F$, say $f$, satisfies the condition (*) $(f(x) - f(y),x - y) \geqslant 0$ for all $x$, $y \in E$. Condition (*) is referred to in the literature as a monotonicity condition. However when $E$ is a Banach lattice, it is natural to define a function $f$ on $E$ into another Banach lattice ${E_1}$ as nondecreasing if $f(x) \geqslant f(y)$ whenever $x$, $y$ in $E$ are such that $x \geqslant y$. The present investigation originated out of the surprising fact that, as shown by suitable examples, the characterization of ${C^1}$-convex real valued functions on the real line recalled above, does not extend if the domain of the function is a Banach lattice of $\dim \geqslant 2$, and the monotonicity of the gradient function is interpreted as in the preceding sentence. More specifically, if $E$ is a Banach lattice of $\dim \geqslant 2$, there are ${C^1}$-convex real valued functions on $E$ with gradients not nondecreasing and there are ${C^1}$-real valued functions on $E$ with gradients nondecreasing which fail to be convex. The main result here provides a complete characterization of ${C^1}$-convex real valued functions $\Phi$ on an arbitrary Banach lattice $E$ of which the translates $\Phi (x + y) - \Phi (x)$ are orthogonally additive in $y$ for all $x$ in $E$, in terms of the monotonicity and natural orthogonality properties of the gradient map of $\Phi$. The characterization of the special class of ${C^1}$-convex functions on Banach lattices stated in the main result here appears to be a reasonable substitute of the characteristic property of ${C^1}$-convex real valued functions on the real line recalled above. The plan of the paper is as follows. In the introductory section, §1, the basic definitions and known results are recalled for convenience of reference, and the main results are stated and proved in §2.

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 first show that if X is a Banach space and α is a left invariant crossnorm on l∞⊗X, then there is a Banach lattice L and an isometric embedding J of X into L, so that I ⊗ J becomes an isometry of l∞⊗αX onto l∞⊗m J(X). Here I denotes the identity operator on l∞ and l∞⊗m J(X) the canonical lattice tensor product. This result is originally due to G. Pisier (unpublished), but our proof is different. We then use this to prove the main results which characterize the Gordon–Lewis property GL and related structures in terms of embeddings into Banach lattices.

In this article, we present a version of martingale theory in terms of Banach lattices. A sequence of contractive positive projections (En) on a Banach lattice F is said to be a filtration if EnEm = En∧m. A sequence (xn )i nF is a martingale if Enxm = xn whenever n m. Denote by M = M(F,(E n)) the Banach space of all norm uniformly bounded martingales. It is shown that if F doesn't contain a copy of c0 or if every En is of finite rank then M is itself a Banach lattice. Convergence of martingales is inves- tigated and a generalization of Doob Convergence Theorem is established. It is proved that under certain conditions one has isometric embeddings F� → M� → F ∗∗ . Finally, it is shown that every martingale difference sequence is a monotone basic sequence. In this paper, we define a martingale in terms of Banach lattices. Rephras- ing the title of (29), this paper could be entitled Martingales without prob- ability. We start with a brief review of classical martingale theory and Banach lattices.

Let \(E\) be a Banach space (or a Banach lattice), let \(\tau\) be a vector topology on \(E\) and let \({\bf x}\) be a sequence (or a positive sequence) in \(E\) not converging to zero with respect to \(\tau\). We show how to construct infinite dimensional Banach spaces (or Banach lattices) consisting, up to the origin, of sequences in \(E\) not converging to zero with respect to \(\tau\) and containing a subsequence of \({\bf x}\). Plenty of applications to Banach space theory and to Banach lattice theory are provided.

Let $T$ be a measurable space, $X$ a Banach space while $Y$ a Banach lattice. We consider the class of ``upper separated'' set-valued functions $F\colon T\times X \rightarrow 2^{Y}$ and investigate the problem of the existence of Caratheodory type selection, that is, measurable in the first variable and order-convex in the second variable.

Our aim is to solve the problem asked by Bahramnezhad and Azar in "KB-operators on Banach lattices and their relationships with Dunford-Pettis and order weakly compact operators". We show that a continuous operators R from a Banach lattice N into a Banach space M is a b-weakly compact operator if and only if R is a KB-operator.

We determine that two recent classes of KB-operators and weak KB-operators and the well-known class of b-weakly compact operators, from a Banach lattice into a Banach space, are the same. We extend our study to the ordered Banach space setting by showing that a weak chain-preserving operator between two ordered Banach spaces is a KB-operator if and only if it is a weak KB-operator.

We prove generalized convergence theorems and Tauberian theorems for vector-valued functions and sequences of growth order γ − 1 with γ > 0 and for positive functions and sequences in Banach lattices. Then the general results are applied to obtain some interesting particular Tauberian results for various examples of operator semigroups. Among them are mean ergodic theorems for Cesaro-mean-bounded semigroups (discrete and continuous) of operators and for semigroups of positive operators.

We use the method of minimal vectors to prove that certain classes of positive quasinilpotent operators on Banach lattices have invariant subspaces. We say that a collection of operators F on a Banach lattice X satisfies condition ( ) if there exists a closed ball B(x0,r) in X such that x0 > 0 and kx0k > r, and for every sequence (xn) in B(x0,r) (0,x0) there exists a subsequence (xni) and a sequence Ki 2 F such that Kixni converges to a non-zero vector. Let Q be a positive quasinilpotent operator on X, one-to-one, with dense range. Denote hQ) = {T > 0 : TQ 6 QT}. If either the set of all operators dominated by Q or the set of all contractions in hQ) satisfies ( ), then hQ) has a common invariant subspace. We also show that if Q is a one-to-one quasinilpotent interval preserving operator on C0(), then hQ) has a common invariant subspace. Lomonosov proved in (Lom73) that if T is not a multiple of the identity and com- mutes with a non-zero compact operator K, then T has a hyperinvariant subspace, that is, a proper closed nontrivial subspace invariant under every operator S in the commutant {T} 0 = {S 2 L(X) : ST = TS}. There has been numerous extensions and generalizations of the result of Lomonosov. In particular, Abramovich, Aliprantis, and Burkinshaw produced several generalizations of Lomonosov's theorem for Banach lattice setting (AAB93, AAB94, AAB98), see also (AA02). In these generalizations commutation relations are substituted by a super-commutation relation ST 6 TS or ST > TS and domination 0 6 K 6 T. They proved a series of results of the fol- lowing type: if S is related to a compact operator via a certain rather loose chain of super-commutations and dominations, then S has an invariant subspace. Ansari and Enflo (AE98) have recently introduced the so-called technique of minimal vectors in order to prove the existence of invariant subspaces for certain classes of operators on a Hilbert space. The method was later modified so that it could be used in arbitrary Banach spaces in (JKP03, And03, CPS04, Tr04). In particular, the method of minimal vectors allows to prove Lomonosov-type results where a compact operator is replaced with a family of operators that mimic a compact operator.

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.

If the vector space \({\mathcal{L}}^r(E, F)\) of all regular operators between the vector lattices E and F is ordered by the collection of its positive operators, then the Dedekind completeness of F is a sufficient condition for \({\mathcal{L}}^r(E, F)\) to be a vector lattice. \({\mathcal{L}}^r(E, F)\) and some of its subspaces might be vector lattices also in a more general situation. In the paper we deal with ordered vector spaces \({\mathcal{V}}(E, F)\) of linear operators and ask under which conditions are they vector lattices, lattice-subspaces of the ordered vector space \({\mathcal{L}}^r(E, F)\) or, in the case that \({\mathcal{L}}^r(E, F)\) is a vector lattice, sublattices or even Banach lattices when equipped with the regular norm. The answer is affirmative for many classes of operators such as compact, weakly compact, regular AM-compact, regular Dunford-Pettis operators and others if acting between appropriate Banach lattices. Then it is possible to study the finite elements in such vector lattices \({\mathcal{V}}(E, F)\), where F is not necessary Dedekind complete. In the last part of the paper there will be considered the question how the order structures of E, F and \({\mathcal{V}}(E, F)\) are mutually related. It is also shown that those rank one and finite rank operators, which are constructed by means of finite elements from E′ and F, are finite elements in \({\mathcal{V}}(E,F)\). The paper contains also some generalization of results obtained for the case \({\mathcal{V}}(E, F) = {\mathcal{L}}^r(E, F)\) in [10].