A generalized mid summability in Banach spaces

Abstract

In this paper, we study the notion of mid summability in a general framework, emphasizing the key role played by the norm iteration property. We introduce the vector valued sequence space λ^mid  (X ) associated with a Banach space X and a sequence space λ. We show that λ^mid  (·) can be placed in a chain with the vector valued sequence spaces λ^s (·) and λ^w (·). Consequently, we define mid λ-summing operators and obtain the maximality of these operator ideals for a suitably restricted sequence space λ. As a result, we obtain a tensor norm corresponding to the maximal ideal of absolutely mid λ-summing operators, using the vector valued sequence spaces.