Ideals of multilinear operators and sequence spaces

CARANDO, DANIEL; DIMANT, VERÓNICA; ROMÁN VILLAFAÑE

Valencia

Workshop; Workshop on functional analysis Valencia 2015; 2015

Universidad de valencia y Universidad politécnica de valencia

\begin{document}\pagestyle{empty}\begin{center}\large\bfseriesIdeals of multilinear operators and sequence spaces\\[0.5\baselineskip]\normalsize\mdseries Daniel Carando, Ver\'onica Dimant, Rom\'an Villafa\~ne\end{center}\noindentLet $E$ and $F$ be Banach sequence spaces. We define, for a sequence $\alpha=(\alpha(k))_{k\in \mathbb{N}}$, the $n$-linear diagonal operator associated with $\alpha$ as\[T_\alpha(x_1,\dots,x_n)=\sum_{k\in\mathbb{N} }\alpha(k)\cdot x_1(k)\cdots x_n(k)\cdot e_k,\]where $x(k)$ denotes the $k$th coordinate of $x\in E$ and $e_k$ denotes the standard unit vector in $c_0$. For a Banach ideal of $n$-linear operators $\mathfrak{A}$, we define the sequence space associated to $\mathfrak A$ as\[\ell_n(\mathfrak{A};E,F)=\left\{\alpha\in\ell_\infty \ : \ T_\alpha\in\mathfrak{A}(^nE,F)\right\}.\]which is a Banach sequence space with the norm $\|\alpha\|_{\ell_n(\mathfrak A;E,F)}:=\|T_\alpha\|_{\mathfrak A(^nE;F)}$.\bigskipIn this talk we study the relationship between the Banach ideals of multilinear operators with their respective associated sequence space. We show some correspondence between a maximal/minimal ideal of multilinear operators and a maximal/minimal sequence space. We consider also the adjoint ideal and derive particular examples.\end{document}