Narrow operators are those operators defined on function spaces which are "small" at signs, i.e., at {-1,0,1}-valued functions. We summarize here some results and problems on them. One of the most interesting things is that if E has an unconditional basis then each operator on E is a sum of two narrow operators, while the sum of two narrow operators on L₁ is narrow. Recently this notion was generalized to vector lattices. This generalization explained the phenomena of sums: the set of all regular narrow operators is a band in the vector lattice of all regular operators (in particular, a subspace). In L₁ all operators are regular, and in spaces with unconditional bases narrow operators with non-narrow sum are non-regular. Nevertheless, a new lattice approach has led to new interesting problems.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc92-0-21,
author = {Mikhail Popov},
title = {Narrow operators (a survey)},
journal = {Banach Center Publications},
volume = {95},
year = {2011},
pages = {299-326},
zbl = {1244.47020},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc92-0-21}
}
Mikhail Popov. Narrow operators (a survey). Banach Center Publications, Tome 95 (2011) pp. 299-326. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc92-0-21/