Domination problems for strictly singular operators and other related classes

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 ).

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“…Another version: if |S| ≤ T , then whether P(T ) implies P(S)? See [2] for a survey on the domination problem for "small" operators.…”

Section: Domination Problem For Abstract Uryson Narrow Operatorsmentioning

confidence: 99%

Narrow orthogonally additive operators

Pliev

Popov

2013

Positivity

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“…Domination properties of positive strictly singular operators on Banach lattices are also studied by J. Flores, F.L. Hernandez and P. Tradacete in [12].…”

Section: Introductionmentioning

confidence: 99%

Domination properties and extension of positive compact operators on pre-Riesz spaces

Gaans¹,

Zhang²

2018

Preprint

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“…Another example of domination between pairs of operators that has been recently studied -in this case, pointwise domination-, can be found in [6] and related references. In this case, the aim is to analyze which properties of the operator T are inherited by the operator S if the relation |S| ≤ |T | holds.…”

Section: Introductionmentioning

confidence: 99%

Maharam-type kernel representation for operators with a trigonometric domination

Pérez

2017

Aequat. Math.

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Consider a linear and continuous operator T between Banach function spaces. We prove that under certain requirements an integral inequality for T is equivalent to a factorization of T through a specific kernel operator: in other words, the operator T has what we call a Maharam-type kernel representation. In the case that the inequality provides a domination involving trigonometric functions, a special factorization through the Fourier operator is given. We apply this result to study the problem that motivates the paper: the approximation of functions in L 2 [0, 1] by means of trigonometric series whose Fourier coefficients are given by weighted trigonometric integrals.

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Domination problems for strictly singular operators and other related classes

Abstract: We survey recent results on domination properties of strictly singular operators and related operator ideals, as well as Banach-Saks operators, Narrow operators and p-summing operators.

Cited by 11 publications

References 46 publications

Narrow orthogonally additive operators

Narrow orthogonally additive operators

Domination properties and extension of positive compact operators on pre-Riesz spaces

Maharam-type kernel representation for operators with a trigonometric domination

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