A generalised mid summability in Banach spaces

TL;DR

This paper introduces a generalized concept of mid summability in Banach spaces using duality theory, defining new sequence spaces and operator ideals, and establishing their relationships and maximality properties.

Contribution

It develops a unified framework for mid summability in Banach spaces, defining new sequence spaces, operator ideals, and tensor norms, extending existing theories.

Findings

Defined vector valued sequence space $ ext{lambda}^{mid}(X)$.

Placed $ ext{lambda}^{mid}(ullet)$ in a chain with $ ext{lambda}^{s}(ullet)$ and $ ext{lambda}^{w}(ullet)$.

Established correspondence between tensor norm and absolutely mid $ ext{lambda}$-summing operators.

Abstract

In this paper, we study the notion of mid summability in a general setting using the duality theory of sequence spaces. We define the vector valued sequence space $\lambda^{mid}(X)$ corresponding to a Banach space $X$ and sequence space $\lambda$. We prove that $\lambda^{mid}(\cdot)$ can be placed in a chain with the vector valued sequence spaces $\lambda^{s}(\cdot)$ and $\lambda^{w}(\cdot)$. Consequently, we define mid $\lambda$-summing operators and obtain the maximality of these operator ideals for a suitably restricted $\lambda$. Furthermore, we define a tensor norm using the vector valued sequence spaces $\lambda^{s}(\cdot)$ and $\lambda^{mid}(\cdot)$, and establish its correspondence with the operator ideal of absolutely mid $\lambda$-summing operators.