Summability properties for the composition of multilinear operators

Source abstract

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 $$\begin{aligned} \Psi _{B}( U_{1},\dots ,U_{n}) =B\hspace{1.111pt}{\circ }\hspace{1.111pt}( U_{1},\dots ,U_{n}), \end{aligned}$$ Ψ B ( U 1 , ⋯ , U n ) = B ∘ ( U 1 , ⋯ , U n ) , and if $$A_{1}:X_{1}\rightarrow Y_{1}$$ A 1 : X 1 → Y 1 , $$\dots$$ ⋯ , $$A_{n}:X_{n}\rightarrow Y_{n}$$ A n : X n → Y n are bounded linear operators let $$\Psi _{\textbf{A},B}:L( Y_{1},Z_{1}) \hspace{1.111pt}{\times }\hspace{1.111pt}\cdots \hspace{1.111pt}{\times }\hspace{1.111pt}L( Y_{n},Z_{n}) \rightarrow \mathscr {A}( X_{1},\dots ,X_{n};W)$$ Ψ A , B : L ( Y 1 , Z 1 ) × ⋯ × L ( Y n , Z n ) → A ( X 1 , ⋯ , X n ; W ) be the composition operator defined by $$\begin{aligned} \Psi _{\textbf{A},B}( U_{1},\dots ,U_{n}) =B\hspace{1.111pt}{\circ }\hspace{1.111pt}( U_{1},\dots ,U_{n}) \hspace{1.111pt}{\circ }\hspace{1.111pt}( A_{1},\dots ,A_{n}) . \end{aligned}$$ Ψ A , B ( U 1 , ⋯ , U n ) = B ∘ ( U 1 , ⋯ , U n ) ∘ ( A 1 , ⋯ , A n ) . We study the summability properties of these compositions of the operators.