Some polynomial versions of cotype and applications

TL;DR

This paper introduces non-linear variants of classical cotype in Banach spaces, explores their properties, and applies them to convergence issues in vector-valued power series and Dirichlet series, also proposing a generalized cotype concept.

Contribution

It develops new non-linear cotype notions for Banach spaces and demonstrates their applications in series convergence and multiplier problems.

Findings

Spaces with l.u.st and cotype satisfy the new non-linear cotype.

Fourier cotype spaces also enjoy the non-linear cotype.

Results on convergence of vector-valued power series and Dirichlet series.

Abstract

We introduce non-linear versions of the classical cotype of Banach spaces. We show that spaces with l.u.st and cotype, and that spaces having Fourier cotype enjoy our non-linear cotype. We apply these concepts to get results on convergence of vector-valued power series in infinite many variables and on $\ell_{1}$-multipliers of vector-valued Dirichlet series. Finally we introduce cotype with respect to indexing sets, an idea that includes our previous definitions.