This is an announcement for the paper "Sur quelques extensions au cadre Banachique de la notion d'op'erateur de Hilbert-Schmidt" by Said Amana Abdillah, Jean Esterle, and Bernhard Hermann Haak.
Abstract: In this work we discuss several ways to extend to the context of Banach spaces the notion of Hilbert-Schmidt operators: $p$-summing operators, $\gamma$-summing or $\gamma$-radonifying operators, weakly $*1$-nuclear operators and classes of operators defined via factorization properties. We introduce the class $PS_2(E; F)$ 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 $H_1$ et $H_2$ 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.
Archive classification: math.FA
Remarks: 18 pages
Submitted from: bernhard.haak@math.u-bordeaux1.fr
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1406.7546
or