A generalised mid summability in Banach spaces

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.