Lattice Sequence Spaces and Summing Mappings

This paper contributes to the theory of positive summing operators between Banach lattices by exploring the interplay between specialized sequence spaces, operator ideals, and tensor product techniques. We focus on the spaces of positive strongly p-summable sequences ℓπp(X) and positive unconditionally p-summable sequences ℓup,|ω|(X), utilizing them alongside the Banach lattice of positive weakly p-summable sequences ℓp,|ω|(X) . These tools are employed to present and characterize three central classes: positive strongly (p,q)-summing operators, positive (p,q)-summing operators, and positive Cohen (p,q)-nuclear operators. Our investigation yields new properties, including the characterization of positive (p,q)-summing operators as those which map positive unconditionally p-summable sequences into q-summable sequences, and the identification of the positive strongly (p,q)-summing class with the class of (p,q)-majorizing operators. A central achievement of this work is the unified characterization of these operator classes via tensor product continuity, a method well-established for linear operator ideals that we now extend to the context of Banach lattices. We characterize each class by the continuity of an associated tensor operator I⊗T:ℓp⊗αX→ℓq⊗βY for appropriate tensor norms α and β. This approach provides a powerful and cohesive framework that deepens the connections between summability, the order structure of Banach lattices, and tensor norms.