An extension of the Maurey Factorization Theorem. Issue 2 (1st February 2020)

• Record Type: Journal Article
• Authors: Popa, Dumitru
• Is Part Of: Linear & multilinear algebra. Volume 68:Issue 2(2020)
• Page Start: 304
• Page End: 322
• Publication Date: 2020-02-01
• Subjects: Maurey factorization theory -- p-summing operators -- almost summing operators -- type -- cotype -- nuclear operators; Primary: 47B38 -- 46E30 -- 46B07 -- Secondary:46B28; Algebras, Linear -- Periodicals; Multilinear algebra -- Periodicals; 512.505
• DOI: 10.1080/03081087.2018.1503632
• Languages: English
• ISSNs: 0308-1087

Abstract

ABSTRACT: Let 1 ≤ p, q, r < ∞ be such that 1 / p = ( 1 / q ) + ( 1 / r ), X, Y Banach spaces, ( Ω, Σ, μ ) a measure space and U : X → L p ( μ, Y ) a bounded linear operator. The Maurey Factorization Theorem gives the necessary and sufficient condition that U can be factored under the form U = M g ∘ V, where V : X → L q ( μ, Y ) is bounded linear and g ∈ L r ( μ ) . In this paper, we define two new classes of bounded linear operators U : X → L p ( μ, Y ) : the class of all operators which satisfies the ( q, p, S ) -Maurey factorization, i.e. there exists g ∈ L r ( μ ) and a ( q, S ) -summing operator V : X → L q ( μ, Y ) such that U = M g ∘ V and the class of ( q, p, S ) -Maurey operators. Our main result, which extends the Maurey Factorization Theorem, asserts that these two classes coincide. Under the assumptions that Y has cotype 2 we prove the equality between the class of operators with values in L p ( μ, Y ) which satisfies the ( 2, p, S ) -Maurey factorization, the class of ( 2, p, S ) -Maurey operators, the class of S -almost summing operators and the class of ( 2, S ) -summing operators. Applications are given.