Let \(1\le p\le \infty \) . By considering a real Banach sequence lattice Y instead of \(\ell ^{p}\) in the definition of p-convexity for a linear operator \(T:E\rightarrow X,\) where E is a Banach space and X a Banach lattice, a natural generalization of p-convexity is obtained, which we call Y-convexity. We show that the space of Y-convex operators from E into X, for which a natural norm is provided, is a Banach space which consists of bounded linear operators and includes those with finite rank. We also discuss the composition of Y-convex operators with bounded linear operators. A study similar to that of Y-convexity is made for Y-concavity and Y-summability and we prove that the composition of a Y-summable linear operator with a bounded linear one is always Y-summable.
