Let $B_w(\ell ^p)$ denote the space of infinite matrices $A$ for which $A(x)\in \ell ^p$ for all $x=\{x_k\}_{k=1}^\infty \in \ell ^p$ with $|x_k|\searrow 0$. We characterize the upper triangular positive matrices from $B_w(\ell ^p)$, $1<p<\infty $, by using a special kind of Schur multipliers and the G. Bennett factorization technique. Also some related results are stated and discussed.