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arXiv · 2004.12325

Nonlinear variants of a theorem of Kwapie\'n

Renato Macedo·Daniel Pellegrino·Joedson Santos
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Abstract

A famous result of S. Kwapie\'{n} asserts that a linear operator from a Banach space to a Hilbert space is absolutely $1$-summing whenever its adjoint is absolutely $q$-summing for some $1\leq q<\infty$; this result was recently extended to Lipschitz operators by Chen and Zheng. In the present paper we show that Kwapie\'{n}'s and Chen--Zheng theorems hold in a very relaxed nonlinear environment, under weaker hypotheses. Even when restricted to the original linear case, our result generalizes Kwapie\'{n}'s theorem because it holds when the adjoint is just almost summing. In addition, a variant for $\mathcal{L}_{p}$-spaces, with $p\geq2$, instead of Hilbert spaces is provided.

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Renato Macedo, Daniel Pellegrino, Joedson Santos. 2020-04-26. Nonlinear variants of a theorem of Kwapie\'n. https://arxiv.org/abs/2004.12325

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