Let a = a n be a sequence of complex numbers. Let d a be the induced multiplier from the Hardy space H p into the sequence space l q where 1 ≤ p q ≤ ∞. We obtain complete characterization for d a to be 2-summing or 1-summing when 1 ≤ p ≤ 2 and 1 ≤ q ≤ ∞, and we give upper and lower estimates for the 2-summing and 1-summing norms in the case 2 < p ≤ ∞ and 1 ≤ q ≤ ∞.

Let α > . By Cα we mean the terraced matrix de ned by c nk = n α if ≤ k ≤ n and if k > n. In this paper, we show that a necessary and su cient condition for the induced operator on l p , to be p-summing, is α > ; ≤ p < ∞. When the more general terraced matrix B, de ned by b nk = βn if ≤ k ≤ n and if k > n, is considered, the necessary and su cient condition turns out to be n n q p * | βn | q < ∞ in the region /p + /q ≤ .

Abstract. We obtain estimates for the distribution of the norm of the random trilinearwhere the a ijk 's are uniformly bounded, independent, mean zero random variables. As an application, we make progress on the problem when r⊗ p⊗ q is a Banach algebra under the Schur multiplication.