Operators with the Maurey–Pietsch multiple splitting property. Issue 6 (2nd June 2020)

• Record Type: Journal Article
• Authors: Popa, Dumitru
• Is Part Of: Linear & multilinear algebra. Volume 68:Issue 6(2020)
• Page Start: 1201
• Page End: 1217
• Publication Date: 2020-06-02
• Subjects: Mikhail Chebotar; p-summing operators -- mixing operators -- p-nuclear operators -- multiple p-summing operators -- Maurey–Pietsch multiple splitting property; Primary: 46G25 -- Secondary: 46B25 -- 46C99; Algebras, Linear -- Periodicals; Multilinear algebra -- Periodicals; 512.505
• DOI: 10.1080/03081087.2018.1536733
• Languages: English
• ISSNs: 0308-1087

Abstract

Abstract : Let n be a natural number, Λ ⊆ N n such that 1, …, 1 ∈ Λ . As a simultaneous extension of the p -summing and multiple p -summing operators we introduce the class of all n -linear operators which are p, Λ -summing, 1 ≤ p < ∞ . If 1 ≤ p ≤ q ≤ ∞, 1 ≤ r ≤ ∞ are such that 1 / p = ( 1 / q ) + ( 1 / r ) we introduce the class M P S q, p Λ of the operators with the Maurey–Pietsch q, p, Λ -splitting property as the multilinear analogue of the class of q, p -mixing linear operators and, as particular classes, M P S q, p, M P S q, p mult . We introduce two new classes of multilinear operators both being the multilinear analogues of the class of r -nuclear operators, namely, the class of r -nuclear operators with respect to the π p Λ -norm, N r π p Λ and the class P r, p Λ and prove that N r ⊂ N r π p Λ ⊂ P r, p Λ ⊂ M P S q, p Λ . By concrete examples, we show that, contrary to the linear case, when Π r ⊂ M q, p, in the multilinear case the inclusions Π 2 mult ⊂ M P 2, 1 mult, Δ 2 ⊂ M P 2, 1 mult are not true, which is in contrast to a well-known result shown by us in another paper that Δ 2 ⊂ M P 2, 1 .