Narrow operators (a survey)

Volume 92 / 2011

Mikhail Popov Banach Center Publications 92 (2011), 299-326 MSC: Primary 46B20; Secondary 46B03, 46B10. DOI: 10.4064/bc92-0-21

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 $L_1$ 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 $L_1$ 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.

Authors

  • Mikhail PopovDepartment of Applied Mathematics
    Chernivtsi National University str. Kotsiubyns'koho 2
    Chernivtsi, 58012 Ukraine
    e-mail

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