IJPAM: Volume 56, No. 1 (2009)

SOME PROPERTIES OF MULTI-LINEAR OPERATORS
F NUCLEAR TYPE

B.M. Cerna
Departamento de Ciencias
Escuela de Matemáticas
Universidad Nacional ``Santiago Antunez de Mayolo"
Ciudad Universitária de Shancayán
100, Avenida Centenario, Huaraz, PERU
e-mail: [email protected]

Abstract.In this work we obtain two results related to multi-linear compact operators in Banach spaces. The first result establishes that an operator $\Phi \in {\cal L}(E_{1},...,E_{n}, F)$ is compact ( $1\leq p \leq \infty$ ) related to $(q_{1}, ...q_{n})$ if and only if there exist the operators $\Psi \in {\cal L}(E_{1},...,E_{n}, l_{p})$ and $A \in {\cal L}(l_{p}, F)$ such that $\Phi = A \,\mbox{o}\, \Psi$ , where $A, \Psi$ are compact. One has $N_{(\infty, p; q_{1}, ...,q_{n})}(\Phi)= inf \vert\vert A\vert\vert \vert\vert\Psi\vert\vert$ , where the infimum is taken over all possible factorizations. The second result is concerned with any multi-linear operator of finite type $\Psi \in {\cal L}(E_{1},...,E_{n}$ , $L_{p}(\Omega, \mu))$ and establishes the following result: