# Narrow operators (a survey)

Banach Center Publications (2011)

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

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topMikhail 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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