We investigate sequences and operators via the unconditionally p-summable sequences. We characterize the unconditionally p-null sequences in terms of a certain tensor product and then prove that, for every 1 ≤ p < ∞, a subset of a Banach space is relatively unconditionally p-compact if and only if it is contained in the closed convex hull of an unconditionally p-null sequence.

Here we study the existence of lower and upper ${\ell }_p$-estimates of sequences in some Banach sequence spaces. We also compute the sharp ${\ell }_p$ estimates in their basis. Finally, we give some applications to weak sequential continuity of polynomials.