Factorization theorems for some new classes of multilinear operators

Two new classes of summing multilinear operators, factorable $(q, p)$-summing operators and $(r; p, q)$-summing operators are studied. These classes are described in terms of factorization. It is shown that operators in the first (resp., the second) class admit the factorization through the injective tensor product of Banach spaces (resp., through some Banach lattices). Applications in different contexts related to Grothendieck Theorem and Fourier integral bilinear operators are shown. Motivated by Pisier’s Theorem on factorization of $(q, 1)$-summing operators from $C(K)$-spaces through Lorentz spaces $L_{q,1}$ on some probability Borel measure spaces, we prove two variants of Pisier’s Theorem for bilinear operators on the product of $C(K)$-spaces. We also prove bilinear versions of Mityagin–Pełczyński and Kislyakov Theorems.

Keywords

bilinear operator, Fourier integral bilinear operators, factorization, Pisier’s Theorem

2010 Mathematics Subject Classification

Primary 46E30. Secondary 46B42, 47B38.

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The first-named author’s research was supported by the National Science Centre of Poland, project 2015/17/B/ST1/00064.

The second-named author’s research was supported by the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigación (Spain) and FEDER under project MTM2016-77054-C2-1-P.

Received 5 January 2018

Accepted 2 April 2019

Published 21 August 2020