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