A way to construct strongly summing and dominated multilinear operators

Let V₁ V 1 , ..., Vₖ V k be non-null bounded multilinear operators, T:Y₁× · · · × Yₖ→ Z T : Y 1 × · · · × Y k → Z a bounded multilinear operator with the property that there exists $$m>0$$ m > 0 such that \| T( y₁,...,yₖ) \| ≥ m\| y₁\| · · · \| yₖ\| T y 1 , . . . , y k ≥ m y 1 · · · y k for all ( y₁,...,yₖ) ∈ Y₁× · · · × Yₖ y 1 , . . . , y k ∈ Y 1 × · · · × Y k and 1≤ p<∞ 1 ≤ p < ∞ . We prove that T∘ ( V₁,...,Vₖ) T ∘ V 1 , . . . , V k is strongly p -summing (resp. p -dominated) if and only if all Vᵢ V i are strongly <jats:i