On C-compact orthogonally additive operators

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Abstract

We consider C-compact orthogonally additive operators in vector lattices. After providing some examples of C-compact orthogonally additive operators on a vector lattice with values in a Banach space we show that the set of those operators is a projection band in the Dedekind complete vector lattice of all regular orthogonally additive operators. In the second part of the article we introduce a new class of vector lattices, called C-complete, and show that any laterally-to-norm continuous C-compact orthogonally additive operator from a C-complete vector lattice to a Banach space is narrow, which generalizes a result of Pliev and Popov.

Introduction

Orthogonally additive operators in vector lattices first were investigated in [12]. Later these results were extended in [1], [2], [6], [7], [8], [18], [21], [22]). Recently, some connections with problems of the convex geometry were revealed [24], [25]. Orthogonally additive operators in lattice-normed spaces were studied in [3]. In this paper we continue this line of research. We analyze the notion of C-compact orthogonally additive operator. In the first part of the article we show that set of all C-compact orthogonally additive operators from a vector lattice E to an order continuous Banach lattice F is a projection band in the vector lattice of all regular orthogonally additive operators from E to F (Theorem 3.9). In the final part of the paper we introduce a new class of vector lattices which we call C-complete (the precise definition is given in section 4) and prove that any laterally-to-norm continuous C-compact orthogonally additive operator from an atomless C-complete vector lattice E to a Banach space X is narrow (Theorem 4.7). This is a generalization of the result of the article [19, Theorem 3.2].

Note that linear narrow operators in function spaces first appeared in [17]. Nowadays the theory of narrow operators is a well-studied object of functional analysis and is presented in many research articles ([9], [11], [15], [14], [19], [22]) and in the monograph [23].

Section snippets

Preliminaries

All necessary information on vector lattices one can find in [4]. In this article all vector lattices are assumed to be Archimedean.

Two elements x,y of a vector lattice E are called disjoint (written as xy), if |x||y|=0. An element a>0 of E is an atom if 0xa,0ya and xy imply that either x=0 or y=0. The equality x=i=1nxi means that x=i=1nxi and xixj for all ij. In the case of n=2 we write x=x1x2. An element y of a vector lattice E is called a fragment (or a component) of xE, if y(x

The projection band of C-compact orthogonally additive operators

In this section we show that the set of all C-compact regular orthogonally additive operators from a vector lattice E to a Banach lattice F with order continuous norm is a band in the vector lattice of all orthogonally additive regular operators from E to F.

Consider some examples.

Example 3.1

Assume that (A,Ξ,μ) and (B,Σ,ν) are σ-finite measure spaces. We say that a map K:A×B×RR is a Carathéodory function if there hold the conditions:

  • (1)

    K(,,r) is μ×ν-measurable for all rR;

  • (2)

    K(s,t,) is continuous on R for μ×

C-compact and narrow orthogonally additive operators

In this section we consider a new class of vector lattices, where the condition of Dedekind completeness is replaced by a much weaker property. For laterally-to-norm continuous, C-compact orthogonally additive operators from a C-complete vector lattice E to a Banach space X we show their narrowness.

Definition 4.1

A vector lattice E is said to be C-complete, if for each xE+ any subset DFx has a supremum and an infimum.

Clearly, every Dedekind complete vector lattice E is C-complete. The reverse statement, in

Acknowledgments

Marat Pliev was supported by the Russian Foundation for Basic Research (grant number 17-51-12064). Martin Weber was supported by the Deutsche Forschungsgemeinschaft (grant number CH 1285/5-1, Order preserving operators in problems of optimal control and in the theory of partial differential equations).

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