IJPAM: Volume 82, No. 1 (2013)

THE MAXIMALITY OF $p$-FACTORABLE
MULTI-LINEAR OPERATORS

B.M. Cerna
Departamento de Ciencias, Escuela de Matemáticas
Universidad Nacional ``Santiago Antunez de Mayolo"
Ciudad Universitária de Shancayán
Avenida Centenario 100, Huaraz, PERU


Abstract. This work demostrate the equivalence of the following definitions ``let $1\leq p\leq \infty,\mbox{ be, }\Phi\in\mathcal{L}(X_{1},\ldots,X_{n};Y)$ is called $p-$factorable, if exist a measure space $(\Omega,\Sigma,\mu)$ and operators $A\in\mathcal{L}(L_{p}(\mu),Y^{**})$ and $\Psi\in\mathcal{L}(X_{1},\ldots,X_{n};L_{p}(\mu))$ such that $K_{Y}\circ\Phi=A\circ\Psi$.

The collection of the $p$-factorable multi-linear operators of $X_{1},\ldots,X_{n}$ to $Y$ will be denoted for $\mathcal{L}_{p\mbox{-\scriptsize {fact}}}(X_{1},\ldots,X_{n},Y)$. Also $\widehat{\gamma}_{p}(\Phi)=\inf\left\Vert\Psi\right\Vert\left\Vert A\right\Vert$, where the infimun is taken over all possible factorzation of $\Phi$ is a norm over $\mathcal{L}_{p\mbox{-\scriptsize {fact}}}(X_{1},\ldots$, $X_{n};Y)$'' and ``let $1\leq p\leq \infty$.

A operator $\Phi\in\mathcal{L}(E_{1},\ldots,E_{n};F)$ is called $p$-factorable relative to $(q_{1},\ldots,q_{n})$ if belongs to the normed ideal:
\begin{multline*} \left[\mathcal{L}_{p\mbox{-\scriptsize {fact}}},\tilde{\gamma}... ...ots; \left[\mathcal{L}_{ap},\left\Vert.\right\Vert\right]^{-1}), \end{multline*}
such that $\frac{1}{p'}+\frac{1}{q'_{1}}+\cdots+\frac{1}{q'_{n}}=1$, with norm

\begin{displaymath}\tilde{\gamma}_{p}(\Phi)=\sup N_{(\infty;p,q_{1},\ldots,q_{n})}(B\Phi(T_{1},\ldots T_{n}))\end{displaymath}

where the supremun is taken on all the $T=\left(T_{1},\ldots,T_{n}\right)$ with $T_{i}\in\mathcal{L}_{ap}\left(X_{i},E_{i}\right)$ and $B\in\mathcal{L}_{ap}(F,F_{0})$ such that $\Vert B\Vert\leq1,\Vert T_{i}\Vert\leq1$, $i=1,\ldots,n$''.

Received: April 17, 2012

AMS Subject Classification: 47H60, 46G25, 47L20

Key Words and Phrases: multi-linear operators, Banach spaces, operator ideals

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2013
Volume: 82
Issue: 1