Narrow Operators on Function Spaces and Vector Lattices

Some problems on narrow operators on function spaces

“…Using the construction from [9] (see also [11,Theorem 5.2]), for every ∈ N the operator I we split as the sum I = Q +Q of two narrow operators Q Q ∈ L(E(Ω )). By Lemma 2.2, the operators R = =1 Q and R = =1 Q are narrow and J = R + R for all ∈ N.…”

“…By definition, the composition of a narrow operator (from the right) by a bounded operator (from the left) is narrow (formally see [11 …”

“…If, moreover, 1 Ω = 1, and for every ∈ L 1 and ∈ E the condition | | = | | implies that ∈ E and = , then the Köthe Banach space E is said to be a rearrangement invariant space (r.i. space, in short) on (Ω Σ µ). It is clear that if E has an absolutely continuous norm then E has an absolutely continuous norm on the unit, however, the converse is not true [11,Example 1.2].…”

“…The notion of a narrow operator generalizes some classes of "small" operators, like compact operators (see [9,11]). However, linear properties of narrow operators are much more involved and not entirely clear.…”

“…To the contrast of narrow operators, the sum of two hereditarily narrow operators is hereditarily narrow [11,Proposition 11.2]. The class of hereditarily narrow operators contains compact operators and some other classes of "small" operators [11,Corollary 11.4].…”