Mid summable sequences: an anisotropic approach

TL;DR

This paper develops a new anisotropic framework for mid summable sequences, introducing the space of mid (q,p)-summable sequences and analyzing associated operator classes with novel theorems.

Contribution

It introduces the space of mid (q,p)-summable sequences and studies the properties of mid (q,p)-summing operators, extending existing theories in a more general setting.

Findings

Established inclusion relations between mid summable sequence spaces.

Defined and analyzed mid (q,p)-summing operators.

Proved new theorems including inclusion, coincidence, and Pietsch Domination for these operators.

Abstract

The notion of mid $p$-summable sequences was introduced by Karn and Sinha in 2014 and recently explored and expanded by Botelho, Campos and Santos in 2017. In this paper we design a theory of mid summable sequences in the anisotropic setting defining a new more general space called space of mid $(q,p)$-summable sequences. As a particular case of our results, we prove an inclusion relation between spaces of mid summable sequences. We also define classes of operators that deals with this new space, the mid $(q,p)$-summing operators, and prove some important results on these classes as inclusion and coincidence theorems and a Pietsch Domination-type theorem. It is worth to mentioning that these abovementioned results are new even in the particular case of the mid $p$-summable environment.