In this work we discuss several ways to extend to the context of Banach spaces the notion of Hilbert-Schmidt operator: p-summing operators, γ-summing or γ-radonifying operators, weakly* 1-nuclear operators and classes of operators defined via factorization properties. We introduce the class PS₂(E;F) of pre-Hilbert-Schmidt operators as the class of all operators u: E → F such that w ∘ u ∘ v is Hilbert-Schmidt for every bounded operator v: H₁ → E and every bounded operator w: F → H₂, where H₁ and H₂ 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm227-3-1,
author = {Said Amana Abdillah and Jean Esterle and Bernhard H. Haak},
title = {Sur quelques extensions au cadre banachique de la notion d'op\'erateur de Hilbert-Schmidt},
journal = {Studia Mathematica},
volume = {231},
year = {2015},
pages = {193-218},
zbl = {06487244},
language = {fra},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm227-3-1}
}
Said Amana Abdillah; Jean Esterle; Bernhard H. Haak. Sur quelques extensions au cadre banachique de la notion d'opérateur de Hilbert-Schmidt. Studia Mathematica, Tome 231 (2015) pp. 193-218. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm227-3-1/