Disjointly non-singular operators: Extensions and local variations

2023/02/09 by Manuel González, González, Manuel, Antonio Martinón +1
Mathematics · #46B42 #47A55 #47B60 #Advanced Banach Space Theory #Advanced Topics in Algebra #Approximation Theory and Sequence Spaces #FOS: Mathematics #Functional Analysis (math.FA)

paper · pdf · doi:10.48550/arxiv.2302.04514

openalex publication_date 2023/02/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The disjointly non-singular (DNS) operators T∈ L(E,Y) from a Banach lattice E to a Banach space Y are those operators which are strictly singular in no closed subspace generated by a disjoint sequence of non-zero vectors. When E is order continuous with a weak unit, E can be represented as a dense ideal in some L₁(μ) space, and we show that each of T∈ DNS(E,Y) admits an extension T∈ DNS(L₁(μ),PO) from which we derive that both T and T^** are tauberian operators and that the operator T^co: E^**/E→ Y^**/Y induced by T^** is an (into) isomorphism. Also, using a local variation of the notion of DNS operator, we show that the ultrapowers of T∈ DNS(E,Y) are also DNS operators. Moreover, when E contains no copies of c₀ and admits a weak unit, we show that T∈ DNS(E,Y) implies T^**∈ DNS(E^**,Y^**).

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