Narrow operators (a survey)
Volume 92 / 2011
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.
