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 Nr(E, F ) of all narrow regular operators is a band in the vector lattice Lr(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 Nr(E, F ) in Lr(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 = TD + TN where TD is a sum of an order absolutely summable family of disjointness preserving operators and TN is narrow. Mathematics Subject Classification (2000). Primary 47B65; secondary 47B38.
Abstract. We extend 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 (2009), pp. 459-495, 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 U lc on (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 U lc n (E, F ).
The paper contains a systematic study of the lateral partial order $$\sqsubseteq $$ ⊑ in a Riesz space (the relation $$x \sqsubseteq y$$ x ⊑ y means that x is a fragment of y) with applications to nonlinear analysis of Riesz spaces. We introduce and study lateral fields, lateral ideals, lateral bands and consistent subsets and show the importance of these notions to the theory of orthogonally additive operators, like ideals and bands are important for linear operators. We prove the existence of a lateral band projection, provide an elegant formula for it and prove some properties of this orthogonally additive operator. One of our main results (Theorem 7.5) asserts that, if D is a lateral field in a Riesz space E with the intersection property, X a vector space and $$T_0:D\rightarrow X$$ T 0 : D → X an orthogonally additive operator, then there exists an orthogonally additive extension $$T:E\rightarrow X$$ T : E → X of $$T_0$$ T 0 . The intersection property of E means that every two-point subset of E has an infimum with respect to the lateral order. In particular, the principal projection property implies the intersection property.
We study the numerical index of absolute sums of Banach spaces, giving general conditions which imply that the numerical index of the sum is less or equal than the infimum of the numerical indices of the summands and we provide some examples where the equality holds covering the already known case of c 0 -, 1 -and ∞ -sums and giving as a new result the case of E-sums where E has the RNP and n(E) = 1 (in particular for finite-dimensional E with n(E) = 1). We also show that the numerical index of a Banach space Z which contains a dense union of increasing one-complemented subspaces is greater or equal than the limit superior of the numerical indices of those subspaces. Using these results, we give a detailed short proof of the already known fact that, for a fixed p, the numerical indices of all infinite-dimensional L p (μ)-spaces coincide.
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