Characterizations of new Cohen summing bilinear operators

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We prove two characterizations of new Cohen summing bilinear operators. The first one is: Let X, Y and Z be Banach spaces, 1 < p < ∞, V : X × Y → Z a bounded linear operator and n ≥ 2 a natural number. Then V is new Cohen p-summing if and only if for all Banach spaces X1, … , Xn and all p-summing operators U : X1 × · · · × Xn → X, the operator V ◦ (U, IY) : X1 × · · · × Xn × Y → Z is -summing. The second result is: Let H be a Hilbert space,, Y, Z Banach spaces and V : H × Y → Z a bounded bilinear operator and 1 < p < ∞. Then V is new Cohen p-summing if and only if for all Banach spaces E and all p-summing operators U : E → H, the operator V ◦ (U, IY) is (p, p*)-dominated.