Sur quelques extensions au cadre banachique de la notion d'opérateur de Hilbert–Schmidt
Volume 227 / 2015
Abstract
In this work we discuss several ways to extend to the context of Banach spaces the notion of Hilbert–Schmidt operator: $p$-summing operators, $\gamma $-summing or $\gamma $-radonifying operators, weakly$^*$ 1-nuclear operators and classes of operators defined via factorization properties. We introduce the class $\mathrm {PS}_2(E; F)$ of pre-Hilbert–Schmidt operators as the class of all operators $u:E\to F$ such that $w\circ u \circ v$ is Hilbert–Schmidt for every bounded operator $v: H_1\to E$ and every bounded operator $w:F\to H_2$, where $H_1$ and $H_2$ are Hilbert spaces. Besides the trivial case where one of the spaces $E$ or $F$ is a ‶Hilbert–Schmidt space″, this space seems to have been described only in the easy situation where one of the spaces $E$ or $F$ is a Hilbert space.
