Majorization for Compact and Weakly Compact Polynomials on Banach Lattices

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

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.

Unbounded order convergence in dual spaces

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 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

The vector space £b(E) of all order bounded linear operators on a Dedekind complete Riesz space E is both a Riesz space and an algebra. This note investigates the degree of compatibility between the algebraic and lattice structures of £b(E). Two of the main results are the following: 1. An operator on a Banach lattice with an order continuous norm factors through the lattice operations if and only if it is an interval preserving Riesz homotnorphism. 2. A Dedekind complete Banach lattice E has an order continuous norm if and only if 0≤Tn ↑ T in £b(E) implies Tn2 ↑ T2.

LetE be a Banach lattice with order continuous norm and {T(t)}t≥0 be an eventually compactc0-semigroup of positive operators onE with generator A. We investigate the structure of the geometric eigenspace of the generator belonging to the spectral bound when the semigroup is ideal reducible. It is shown that a basis of the eigenspace can be chosen to consist of element ofE with certain positivity structure. This is achieved by a decomposition of the underlying Banach latticeE into a direct sum of closed ideals which can be viewed as a generalization of the Frobenius normal form for nonnegative reducible matrices.

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 .

We show that even in the case of a Banach lattice E with an order continuous norm (or whose dual has an order continuous norm) the linear span of the positive compact operators on E need not be complete under the regular norm.

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

We establish the domination property and some lattice approximation properties for almost L-weakly and almost M-weakly compact operators. Then, we consider the linear span of positive almost L-weakly (resp., almost M-weakly) compact operators and give results about when they form a Banach lattice and have an order continuous norm.

Some results on the class of U-Dunford-Pettis operators

Of concern is the unbounded operatorAΦf=f′{A_\Phi }f = f’with nonlinear domainD(AΦ)={f∈W1,1:f(0)=Φ(f)}D({A_\Phi }) = \{ f \in {W^{1,1}}:f(0) = \Phi (f)\}which is considered on the Banach spaceEEof Bochner integrable functions on an interval with values in a Banach spaceFF. Under the assumption thatΦ\Phiis a Lipschitz continuous operator fromEEtoFF, it is shown thatAΦ{A_{\Phi }}generates a strongly continuous translation semigroup(TΦ(t))t≥0{({T_\Phi }(t))_{t \geq 0}}. For linear operatorsΦ\Phiseveral properties such as essential-compactness, positivity, and irreducibility of the semigroup(TΦ(t))t≥0{({T_\Phi }(t))_{t \geq 0}}depending on the operatorΦ\Phiare studied. It is shown that ifFFis a Banach lattice with order continuous norm, then(TΦ(t))t≥0{({T_\Phi }(t))_{t \geq 0}}is the modulus semigroup of(TΦ(t))t≥0{({T_\Phi }(t))_{t \geq 0}}. Finally spectral properties ofAΦ{A_\Phi }are studied and the spectral bounds(AΦ)s({A_\Phi })is determined. This leads to a result on the global asymptotic behavior in the case whereΦ\Phiis linear and to a local stability result in the case whereΦ\Phiis Fréchet differentiable.