# Narrow operators (a survey)

• Volume: 92, Issue: 1, page 299-326
• ISSN: 0137-6934

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## Abstract

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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.

## How to cite

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Mikhail Popov. "Narrow operators (a survey)." Banach Center Publications 92.1 (2011): 299-326. <http://eudml.org/doc/281998>.

@article{MikhailPopov2011,
abstract = {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.},
author = {Mikhail Popov},
journal = {Banach Center Publications},
keywords = {vector lattice; band; symmetric Banach space; absolutely continuous norm; complemented subspace; order completeness of a vector lattice; unconditional basis; narrow operator; hereditarily narrow operator; Dunford-Pettis operator; weakly compact operator; numerical radius; Daugavet property},
language = {eng},
number = {1},
pages = {299-326},
title = {Narrow operators (a survey)},
url = {http://eudml.org/doc/281998},
volume = {92},
year = {2011},
}

TY - JOUR
AU - Mikhail Popov
TI - Narrow operators (a survey)
JO - Banach Center Publications
PY - 2011
VL - 92
IS - 1
SP - 299
EP - 326
AB - 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.
LA - eng
KW - vector lattice; band; symmetric Banach space; absolutely continuous norm; complemented subspace; order completeness of a vector lattice; unconditional basis; narrow operator; hereditarily narrow operator; Dunford-Pettis operator; weakly compact operator; numerical radius; Daugavet property
UR - http://eudml.org/doc/281998
ER -

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