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Some problems on narrow operators on function spaces. (English) Zbl 1293.46013

Let \(p_n \in (0, 2)\), \(p_n \to 1\), and let \([0, 1] = \bigcup_n \Omega_n\) be a disjoint partition into subsets of positive measure. It is shown that the \(\ell_2\)-direct sum \(E\) of the spaces \(L_{p_n}(\Omega_n)\) has the following two properties: (i) \(E\) cannot be embedded in a Banach space with an unconditional basis, and (ii) operators \(T: E \to E\) that are decomposable as a sum of two narrow operators form a dense subset in \(L(E)\) in the sense of point-wise convergence. It remains an open question whether there exists a rearrangement invariant space \(E\) on \([0, 1] \) with such properties. Another result demonstrates, in particular, that the orthogonal projection of \(L_p[0,1]\), \(2 < p < \infty\), onto the closed linear span of the Rademacher system is a hereditarily narrow operator. This answers Problem 11.9 of M. Popov and B. Randrianantoanina [Narrow operators on function spaces and vector lattices. de Gruyter Studies in Mathematics 45. Berlin: de Gruyter (2013; Zbl 1258.47002)]. Several open questions are formulated.
For the definitions of narrow and hereditarily narrow operators, we refer to the recent book of Popov and Randrianantoanina cited above.

MSC:

46B42 Banach lattices
47B38 Linear operators on function spaces (general)

Citations:

Zbl 1258.47002
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References:

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