Ideal structure and factorization properties of the regular kernel operators

A. Blanco*

*Corresponding author for this work

Research output: Contribution to journalArticle

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Abstract

We consider the structure of the lattice of (order and algebra) ideals of the band of regular kernel operators on Lp-spaces. We show, in particular, that for any Lp(μ) space, with μσ-finite and 1 < p< ∞, the norm-closure of the ideal of finite-rank operators on Lp(μ) , is the only non-trivial proper closed (order and algebra) ideal of this band. Key to our results in the Lp setting is the fact that every regular kernel operator on an Lp(μ) space (μ and p as before) factors with regular factors through ℓp. We show that a similar but weaker factorization property, where ℓp is replaced by some reflexive purely atomic Banach lattice, characterizes the regular kernel operators from a reflexive Banach lattice with weak order unit to a KB-space with weak order unit.

Original languageEnglish
Number of pages19
JournalPositivity
Early online date18 Dec 2019
DOIs
Publication statusEarly online date - 18 Dec 2019

Keywords

  • Algebra ideal
  • Banach lattice
  • Kernel operator
  • Order ideal
  • Regular operator

ASJC Scopus subject areas

  • Analysis
  • Theoretical Computer Science
  • Mathematics(all)

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