Invariant sublattices for positive operators

There are, by now, many results which guarantee that positive operators on Banach lattices have non-trivial closed invariant sublattices. In particular, this is true for every positive compact operator. Apart from some results of a general nature, in this paper we present several examples of positive operators on Banach lattices which do not have non-trivial closed invariant sublattices. These examples include both AM-spaces and Banach lattices with an order continuous norm and which are and are not atomic. In all these cases we can ensure that the operators do possess non-trivial closed invariant subspaces.     

A characterization of dual Banach lattices

In this paper we give a characterization of dual Banach lattices. In fact, we prove that a Banach function space E on a separable measure space which has the Fatou property is a dual Banach lattice if and only if all positive operators from L¹(0,1) into E are abstract kernel operators, hence extending the fact, proved by M. Talagrand, that separable Banach lattices with the Radon-Nikodym property are dual Banach lattices.     

ON BAND IRREDUCIBLE OPERATORS ON BANACH LATTICES

Abstract

In this note we prove an extension of B. de Pagter's theorem (positive, compact, irreducible operators on Banach lattices are not quasi-nilpotent) for positive, band irreducible operators on Banach lattices satisfying some additional conditions.     

Some Results on b-AM-Compact Operators

We characterize Banach lattices for which the class of order weakly compact operators (resp. b-weakly compact operators) between Banach lattices coincide with that of b-AM-compact operators.     

Some Applications of Rademacher Sequences in Banach Lattices

We give several applications of Rademacher sequences in abstract Banach lattices. We characterise those Banach lattices with an atomic dual in terms of weak* sequential convergence. We give an alternative treatment of results of Rosenthal, generalising a classical result of Pitt, on the compactness of operators from L^p into L^q. Finally we generalise earlier work of ours by showing that, amongst Banach lattices F with an order continuous norm, those having the property that the linear span of the positive compact operators fromE into F is complete under the regular norm for all Banach lattices E are precisely the atomic lattices.     

r-ASYMPTOTICALLY QUASI-FINITE RANK OPERATORS AND THE SPECTRUM OF MEASURES

Abstract

The r-asymptotically quasi finite rank operators were introduced in [10]. For regular operators on Banach lattices, these operators are the order theoretic analogue of Riesz operators on Banach spaces. We establish their basic properties and apply these in the spectral analysis of convolution operators.     

The multisublinear maximal type operators in Banach function lattices

The main aim of this paper is to study a general multisublinear operators generated by quasi-concave functions between weighted Banach function lattices. These operators, in particular, generalize the Hardy–Littlewood and fractional maximal functions playing an important role in harmonic analysis. We prove that under some general geometrical assumptions on Banach function lattices two-weight weak type and also strong type estimates for these operators are true. To derive the main results of this paper we characterize the strong type estimate for a variant of multilinear averaging operators. As special cases we provide boundedness results for fractional maximal operators in concrete function spaces.     

Composition and nuclearity of kernel operators

Let T and S be kernel operators of finite double norm, respectively finite inverse double norm. We prove that the compositions TS and ST are nuclear operators, provided one of the function norms is order continuous. Conversely we prove theorems about factorizations of nuclear kernel operators on Banach function spaces. These results are then used to derive optimal eigenvalue estimates for nuclear operators on Banach lattices by relating Pisier's results with the results of König and Weis.     

On the weak compactness of b-weakly compact operators

We characterize Banach lattices on which the class of b-weakly compact operators coincides with that of weakly compact operators.     

Spectral properties of disjointly strictly singular operators

Spectral properties of strictly singular and disjointly strictly singular operators on Banach lattices are studied. We show that even in the case of positive operators, the whole spectral theory of strictly singular operators cannot be extended to disjointly strictly singular operators. However, several spectral properties of disjointly strictly singular operators are given.     

The b-Weak Compactness of Order Weakly Compact Operators

We characterize Banach lattices on which the class of order weakly compact operators coincides with that of b-weakly compact operators.     

Order Structures in Banach Spaces of Analytic Functions

In this paper we consider some Banach spaces of analytic functions on the unit disk generated by the cone of analytic functions with monotone decreasing Taylor coefficients. We get that some of these spaces are Banach lattices with respect to this cone. Different ordered spaces of linear bounded operators acting between the previous spaces are also investigated, with emphasis on the so-called regular multipliers and Hankel operators.     

Polynomials on Banach lattices and positive tensor products

This paper is the first systematic study of homogeneous polynomials on Banach lattices. A variety of new Banach spaces and Banach lattices of multilinear maps, homogeneous polynomials, and operators are introduced. The main technique is to employ positive tensor products and quotients of positive tensor products. Our theorems generalize the results on orthogonally additive polynomials by Benyamini, Lassalle, and Llavona (2006) in [4], the results by Grecu and Ryan (2005) in [14], and the results by Sundaresan (1991) in [23].     

Strictly singular and power-compact operators on Banach lattices

Compactness of the iterates of strictly singular operators on Banach lattices is analyzed. We provide suitable conditions on the behavior of disjoint sequences in a Banach lattice, for strictly singular operators to be Dunford-Pettis, compact or have compact square. Special emphasis is given to the class of rearrangement invariant function spaces (in particular, Orlicz and Lorentz spaces). Moreover, examples of rearrangement invariant function spaces of fixed arbitrary indices with strictly singular non power-compact operators are also presented.     

About positive weak Dunford-Pettis operators on Banach lattices

We characterize Banach lattices for which each positive weak Dunford-Pettis operator from a Banach lattice into another dual Banach lattice is almost Dunford-Pettis. Also, we give some sufficient and necessary conditions for which the class of positive weak Dunford-Pettis operators coincides with that of positive Dunford-Pettis operators, and we derive some consequences.     

Schur operators and domination problem

In this paper we are concerned with developing generalizing concepts of Dunford–Pettis operators analogous to the generalization of compact operators by strictly singular operators. Also, we give some new results concerning the domination problem in the setting of positive operators between Banach lattices.     

L-weakly and M-weakly compact operators and the centre

We extend known results concerning the centre of spaces of regular (resp. weakly compact or compact) operators between two Banach lattices to the setting of L-weakly compact and M-weakly compact operators. We also show that the L-weakly compact, M-weakly compact, and compact operators lying in the centre of a Banach lattice coincide.     

Some characterizations of compact operators on Banach lattices

We study the compactness of the class of operators which are AM-compact and semi-compact on Banach lattices and as consequences, we obtain some characterizations of order continuous norms.      

The duality problem for the Class of b-weakly compact operators

We establish necessary and sufficient conditions under which b-weakly compact operators between Banach lattices have b-weakly compact adjoints or operators with b-weakly compact adjoints are themselves b-weakly compact. Also, we give some consequences.      

About b-weakly compact operators on Banach lattices

We characterize Banach lattices on which each positive operators is b-weakly compact and we derive some characterizations of KB-spaces.