Duality between Y-convexity and $Y^{\times }$-concavity of linear operators between Banach lattices

In this paper we study the Y-convexity, a property which is obtained by considering a real Banach sequence lattice Y instead of ${\mathrm{\ell }}^p$ for a linear operator $T:E\to X$, where E is a Banach space and X is a Banach lattice. We introduce some vector sequence spaces in order to characterize the Y-convexity of T by means of the continuity of an associated operator $\bar{T}$. Analogous results for Y-concavity are also obtained. Finally, the duality between Y-convexity and $Y^{\times }$-concavity is proven.