Summability properties for the composition of multilinear operators

Abstract Let $$\mathscr {A}$$ A be a Banach ideal of bounded multilinear operators. If $$ A\in \mathscr {A}( X_{1},\dots ,X_{n};Y) $$ A ∈ A ( X 1 , ⋯ , X n ; Y ) we consider the composition operator $$\Theta _{A}:L( Y,Z) \rightarrow \mathscr {A} ( X_{1},\dots ,X_{n};Z) $$ Θ A : L ( Y , Z ) → A ( X 1 , ⋯ , X n ; Z ) defined by $$\Theta _{A}( U) =U\hspace{1.111pt}{\circ }\hspace{1.111pt}A$$ Θ A ( U ) = U ∘ A . Similarly, if $$B\in \mathscr {A}( Z_{1},\dots ,Z_{n};W) $$ B ∈ A ( Z 1 , ⋯ , Z n ; W ) let $$\Psi _{B}:L( Y_{1},Z_{1}) \hspace{1.111pt}{\times }\hspace{1.111pt}\cdots \hspace{1.111pt}{\times }\hspace{1.111pt}L( Y_{n},Z_{n}) \rightarrow \mathscr {A}( Y_{1},\dots ,Y_{n};W) $$ Ψ B : L ( Y 1 , Z 1 ) × ⋯ × L ( Y n , Z n ) → A ( Y 1 , ⋯ , Y n ; W ) be the composition operator defined by $$\begin{aligned} \Psi _{B}( U_{1},\dots ,U_{n}) =B\hspace{1.111pt}{\circ }\hspace{1.111pt}(