A lattice approach to narrow 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 N_r(E, F) of all narrow regular operators is a band in the vector lattice L_r(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 preserving operators is the orthogonal complement to N_r(E, F) in L_r(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 = T_D + T_N where T_D is a sum of an order absolutely summable family of disjointness preserving operators and T_N is narrow. Supported by Ukr. Derzh. Tema N 0103Y001103.     

Domination problem for narrow orthogonally additive operators

The “Up-and-down” theorem which describes the structure of the Boolean algebra of fragments of a linear positive operator is the well known result in operator theory. We prove an analog of this theorem for a positive abstract Uryson operator defined on a vector lattice and taking values in a Dedekind complete vector lattice. This result is used to prove a theorem of domination for order narrow positive abstract Uryson operators from a vector lattice E to a Banach lattice F with an order continuous norm.     

Narrow orthogonally additive operators

We generalize the notion of narrow operators to nonlinear maps on vector lattices. The main objects are orthogonally additive operators and, in particular, abstract Uryson operators. Most of the results extend known theorems obtained by O. Maslyuchenko, V. Mykhaylyuk and the second named author published in Positivity 13:459–495 (2009) for linear operators. For instance, we prove that every orthogonally additive laterally-to-norm continuous C-compact operator from an atomless Dedekind complete vector lattice to a Banach space is narrow. Another result asserts that the set \({\mathcal U}_{on}^{lc}(E,F)\) of all order narrow laterally continuous abstract Uryson operators is a band in the vector lattice of all laterally continuous abstract Uryson operators from an atomless vector lattice \(E\) with the principal projection property to a Dedekind complete vector lattice \(F\) . The band generated by the disjointness preserving laterally continuous abstract Uryson operators is the orthogonal complement to \({\mathcal U}_n^{lc}(E,F)\) .     

Domination by Positive Narrow Operators

We prove that each positive operator from a Köthe function-space E() to a Banach lattice F with a narrow majorant is itself narrow provided the norm on F is order continuous. We also prove that every l ²-strictly singular regular operator from L ^p[0,1], 1p < , to a Banach lattice F is narrow, provided F has an order continuous norm.     

On order structure and operators in L ∞(μ)

It is known that there is a continuous linear functional on L _∞ which is not narrow. On the other hand, every order-to-norm continuous AM-compact operator from L _∞(μ) to a Banach space is narrow. We study order-to-norm continuous operators acting from L _∞(μ) with a finite atomless measure μ to a Banach space. One of our main results asserts that every order-to-norm continuous operator from L _∞(μ) to c ₀(Γ) is narrow while not every such an operator is AM-compact.     

Multi-hypercyclic operators are hypercyclic

Herrero conjectured in 1991 that every multi-hypercyclic (respectively, multi-supercyclic) operator on a Hilbert space is in fact hypercyclic (respectively, supercyclic). In this article we settle this conjecture in the affirmative even for continuous linear operators defined on arbitrary locally convex spaces. More precisely, we show that, if is a continuous linear operator on a locally convex space E such that there is a finite collection of orbits of T satisfying that each element in E can be arbitrarily approximated by a vector of one of these orbits, then there is a single orbit dense in E. We also prove the corresponding result for a weaker notion of approximation, called supercyclicity . Received October 18, 1999 / Published online February 5, 2001     

On the “function” and “lattice” definitions of a narrow operator

We prove that the function and lattice definitions of a narrow operator defined on a Köthe Banach space E on a finite atomless measure space \((\Omega , \Sigma , \mu )\) are equivalent if and only if the set of all simple functions is dense in E. This answers Problem 10.3 from Popov and Randrianantoanina (Narrow operators on function spaces and vector lattices, De Gruyter studies in mathematics 45, De Gruyter, Berlin, 2013).     

On lattice properties of the composition 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≤T_n ↑ T in £_b(E) implies T _n ² ↑ T².

     

A Remark on the Local Lipschitz Continuity of Vector Hysteresis Operators

It is known that the vector stop operator with a convex closed characteristic Z of class C ¹ is locally Lipschitz continuous in the space of absolutely continuous functions if the unit outward normal mapping n is Lipschitz continuous on the boundary Z of Z. We prove that in the regular case, this condition is also necessary.     

Sublattices of complete lattices with continuity conditions

Various embedding problems of lattices into complete lattices are solved. We prove that for any join-semilattice S with the minimal join-cover refinement property, the ideal lattice Id S of S is both algebraic and dually algebraic. Furthermore, if there are no infinite D-sequences in J(S), then Id S can be embedded into a direct product of finite lower bounded lattices. We also find a system of infinitary identities that characterize sublattices of complete, lower continuous, and join-semidistributive lattices. These conditions are satisfied by any (not necessarily finitely generated) lower bounded lattice and by any locally finite, join-semidistributive lattice. Furthermore, they imply M. Erné’s dual staircase distributivity.On the other hand, we prove that the subspace lattice of any infinite-dimensional vector space cannot be embedded into any ℵ₀-complete, ℵ₀-upper continuous, and ℵ₀-lower continuous lattice. A similar result holds for the lattice of all order-convex subsets of any infinite chain.Dedicated to the memory of Ivan RivalReceived April 4, 2003; accepted in final form June 16, 2004.This revised version was published online in August 2005 with a corrected cover date.     

A Note on <Emphasis Type="Italic">b</Emphasis>-Weakly Compact Operators

We consider a continuous operator T: E → X where E is a Banach lattice and X is a Banach space. We characterize the b-weak compactness of T in terms of its mapping properties.     

On sums of narrow and compact operators

We prove, in particular, that if E is a Dedekind complete atomless Riesz space and X is a Banach space then the sum of a narrow and a C-compact laterally continuous orthogonally additive operators from E to X is narrow. This generalizes in several directions known results on narrowness of the sum of a narrow and a compact operators for the settings of linear and orthogonally additive operators defined on Köthe function spaces and Riesz spaces.

     

Narrow orthogonally additive operators in lattice-normed spaces

We consider a new class of narrow orthogonally additive operators in lattice-normed spaces and prove the narrowness of every C-compact norm-laterally-continuous orthogonally additive operator from a Banach–Kantorovich space V into a Banach space Y. Furthermore, every dominated Urysohn operator from V into a Banach sequence lattice Y is also narrow. We establish that the order narrowness of a dominated Urysohn operator from a Banach–Kantorovich space V into a Banach space with mixed norm W implies the order narrowness of the least dominant of the operator.     

On extension of abstract Urysohn operators

We consider the extension of an orthogonally additive operator from a lateral ideal and a lateral band to the whole space. We prove in particular that every orthogonally additive operator, extended from a lateral band of an order complete vector lattice, preserves lateral continuity, narrowness, compactness, and disjointness preservation. These results involve the strengthening of a recent theorem about narrow orthogonally additive operators in vector lattices.     

Summability and vector amarts

We show that an operator is absolutely summing if and only if it maps amarts into uniform amarts, from which we can deduce a theorem of A. Bellow and another of Edgar-Sucheston. We also show that the absolute value of a Banach lattice valued potential is a potential if and only if the lattice is an A-M space from which we deduce that the L¹-bounded amarts form a Ricsz space if and only if the space is finite dimensional.     

On the exponential dichotomy and spectral properties of difference operators related to the Howland semigroup

We consider the linear difference operator (Dx)(t) = x(t) − B(t)x(t − h), t ∈ ℝ, where h ∈ ℝ is a nonzero number and B is a continuous bounded operator-valued function on ℝ ranging in the algebra of linear bounded operators on a Banach space. We obtain necessary and sufficient conditions for the continuous invertibility of D in the space of continuous bounded vector functions and present a formula for the inverse operator.     

On Prebox Module Topologies

Let R be a complete topological division ring whose topology is determined by a real-valued valuation, and let M be a vector space over R. It is proved that M admits a Hausdorff module topology preceding the box topology in the lattice of all module topologies if and only if the dimension of the vector space M over R is a measurable cardinal.     

On Finite Elements in Lattices of Regular Operators

Let E and F be vector lattices and the ordered space of all regular operators, which turns out to be a (Dedekind complete) vector lattice if F is Dedekind complete. We show that every lattice isomorphism from E onto F is a finite element in , and that if E is an AL-space and F is a Dedekind complete AM-space with an order unit, then each regular operator is a finite element in . We also investigate the finiteness of finite rank operators in Banach lattices. In particular, we give necessary and sufficient conditions for rank one operators to be finite elements in the vector lattice . A half year stay at the Technical University of Dresden was supported by China Scholarship Council.     

Orthomodular lattices in ordered vector spaces

In a partly ordered space the orthogonality relation is defined by incomparability. We define integrally open and integrally semi-open ordered real vector spaces. We prove: if an ordered real vector space is integrally semi-open, then a complete lattice of double orthoclosed sets is orthomodular. An integrally open concept is closely related to an open set in the Euclidean topology in a finite dimensional ordered vector space. We prove: if V is an ordered Euclidean space, then V is integrally open and directed (and is also Archimedean) if and only if its positive cone, without vertex 0, is an open set in the Euclidean topology (and also the family of all order segments , a < b, is a base for the Euclidean topology). Received January 7, 2005; accepted in final form November 26, 2005.     

Continuity properties of Markov semigroups and their restrictions to invariant <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-spaces

We consider Markov semigroups on the cone of positive finite measures on a complete separable metric space. Such a semigroup extends to a semigroup of linear operators on the vector space of measures that typically fails to be strongly continuous for the total variation norm. First we characterise when the restriction of a Markov semigroup to an invariant L ¹-space is strongly continuous. Aided by this result we provide several characterisations of the subspace of strong continuity for the total variation norm. We prove that this subspace is a projection band in the Banach lattice of finite measures, and consequently obtain a direct sum decomposition.