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Let B _w (ℓ ^p ) denote the space of infinite matrices A for which A(x) ∈ ℓ ^p for all x = {x _k } _k=1 ^∞ ∈ ℓ ^p with |x _k | ↘ 0. We characterize the upper triangular positive matrices from B _w (ℓ ^p ), 1 < p < ∞, by using a special kind of Schur multipliers and the G. Bennett factorization technique. Also some related results are stated and discussed.