On C-compact orthogonally additive operators
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 of a vector lattice E are called disjoint (written as ), if . An element of E is an atom if and imply that either or . The equality means that and for all . In the case of we write . An element y of a vector lattice E is called a fragment (or a component) of , if
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 and are σ-finite measure spaces. We say that a map is a Carathéodory function if there hold the conditions: is -measurable for all ; is continuous on for
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 any subset 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).
References (26)
- et al.
On extensions of some nonlinear maps in vector lattices
J. Math. Anal. Appl.
(2017) Lattice preserving maps on lattices of continuous functions
J. Math. Anal. Appl.
(2013)A factorization for orthogonally additive operators on Banach lattices
J. Math. Anal. Appl.
(2019)On the sum of narrow and a compact operators
J. Funct. Anal.
(2014)- et al.
On sums of narrow operators on Köthe function space
J. Math. Anal. Appl.
(2013) - et al.
Continuity and representation of valuations on star bodies
Adv. Math.
(2018) - et al.
Disjointness preserving orthogonally additive operators in vector lattices
Banach Math. Anal.
(2018) - et al.
Dominated orthogonally additive operators in lattice-normed spaces
Adv. Oper. Theory
(2019) - et al.
Positive Operators
(2006) - et al.
Nonlinear Superposition Operators
(2008)