  In this paper, considering a real Banach sequence lattice Y instead of a
Lebesgue sequence space $l_p$ we generalize p-convexity of a linear operator
$T:E\to X$, where E is a Banach space and X is a Banach lattice. Then we prove
that basic properties of p-convexity remain valid for Y-convex linear
operators. Analogous generalizations are given for q-concavity and
p-summability and composition properties between these operators are analyzed.
