A. K. Kitover

A solution to a problem on invertible disjointness preserving operators

We construct an invertible disjointness preserving operator T on a normed lattice such that T−1 is not disjointness preserving. Two elements x1, x2 in an (Archimedean) vector lattice X are said to be disjoint (in symbols: x1 ⊥ x2) if |x1| ∧ |x2| = 0. Recall that a (linear) operator T : X → Y between vector lattices is said to be disjointness preserving if T sends any two disjoint elements in X to disjoint elements in Y, that is, Tx1 ⊥ Tx2 in Y whenever x1 ⊥ x2 in X . Assume now that a disjointness preserving operator T : X → Y is bijective, i.e., one-to-one and onto, so that T−1 : Y → X exists. It was conjectured by the first named author (see, for example, the problem section in [HL, page 143]) that T−1 is also disjointness preserving. The purpose of this note is to give a counterexample to this conjecture. Moreover, we will construct a bijective disjointness preserving automorphism T on a normed lattice G such that T−1 is not disjointness preserving. We precede our construction by several comments on some recent developments regarding the above conjecture. The two most important classes of vector lattices are the Banach lattices and the Dedekind complete vector lattices. The vector lattice G that we will construct is in neither of these classes. It cannot be a Banach lattice in view of the fact that, as has been recently shown by Huijsmans and de Pagter [HP] and independently by Koldunov [K], the above conjecture has an affirmative solution for Banach lattices. More precisely, they proved the following theorem. Theorem 1 ([HP, Theorem 2.1], [K, Theorem 3.6]). Let T be a one-to-one disjointness preserving operator from a relatively uniformly complete vector lattice X into a normed lattice Y . Then T is a d-isomorphism, that is, x1 ⊥ x2 in X if and only if Tx1 ⊥ Tx2 in Y . In particular, if TX = Y then T−1 : Y → X is a disjointness preserving operator. Since each Banach lattice is relatively uniformly complete, the previous theorem includes the case of Banach lattices. Here we should also cite the paper by K. Jarosz [J], which preceded [HP] and [K] and in which a special case of Theorem 1 was Received by the editors November 7, 1996. 1991 Mathematics Subject Classification. Primary 47B60.

Invariant sublattices for positive operators

Abstract 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 Operators Preserving Disjointness in Terms of their Inverse

The characterization mentioned in the title is found.

Spectrum of Weighted Composition Operators Part III: Essential Spectra of Some Disjointness Preserving Operators on Banach Lattices

We describe essential (in particular Fredholm and semi-Fredholm) spectra of operators on Banach lattices of the form T = wU, where w is a central operator and U is a disjointness preserving operator such that its spectrum σ(U) is a subset of the unit circle.

Spectrum of operators in ideal spaces

One considers “weighted translation” operators in ideal Banach spaces. It is proved that if the translation is aperiodic (the set of periodic points has measure zero), then the spectrum of such an operator is rotationinvariant. This result can be extended (under certain additional restrictions) to “weighted translation” operators acting in regular subspaces of ideal spaces, in particular, to operators in Hardy spaces.In this note we prove the rotation-invariance of the spectrum of aperiodic operators of “weighted translation” in ideal spaces and uniform B-algebras.

The dual Radon–Nikodym property for finitely generated Banach C ( K )-modules

We extend the well-known criterion of Lotz for the dual Radon–Nikodym property (RNP) of Banach lattices to finitely generated Banach C(K)-modules and Banach C(K)-modules of finite multiplicity. Namely, we prove that if X is a Banach space from one of these classes then its Banach dual

Bijective Disjointness Preserving Operators

X^\star

Inverses of disjointness preserving operators

X⋆ has the RNP iff X does not contain a closed subspace isomorphic to