Domination by positive narrow operators
Let \(E (\mu)\) be a Köthe function space on a probability space \((\Omega, \Sigma, \mu)\) with an atomless measure \(\mu\) and \(Y\) be a Banach space. An operator \(T: E (\mu) \to Y\) is called narrow if for each measurable set \(A \in \Sigma\) and very \(\varepsilon > 0\) there is a partition \(\{A_1, A_2\}\) of \(A\) with \(\mu (A_1) = \mu (A_2)\) with \(\| T (\chi_{A_1} - \chi_{A_2}) \| < \varepsilon\). \textit{A. M. Plichko} and \textit{M. M. Popov} asked the following question in [``Symmetric function spaces on atomless probability spaces'' (Diss. Math. 306) (1990; Zbl 0715.46011)]: Let \(1 < p < \infty\), \(p \neq 2\) and \(Y\) a Banach space. Is every strictly singular operator \(T: L^p (\Omega, \Sigma, \mu) \to Y\) narrow? Is every \(\ell^2\)-singular operator \(T: L^p (\Omega, \Sigma, \mu) \to Y\) narrow? The authors answer this question in the setting of regular operators between Banach lattices. More precisely, they obtain the following Theorem: Let \(F\) be an order continuous Banach lattice and \((\Omega, \Sigma, \mu)\) be a probability space with an atomless measure \(\mu\). Let \(T: L^p (\Omega, \Sigma, \mu) \to F\) be a regular operator, \(1 \leq p \leq \infty\). If \(T\) is \(\ell^2\)-singular, then \(T\) is narrow. Among other things, they note that the converse of this theorem is not true (Example 2.8). The authors also study the problem of domination by positive narrow operators. That is, if \(F\) is a Banach lattice and \(0 \leq S \leq T\): \(E (\mu) \to F\) are two positive operators with \(T\) narrow, which conditions on \(E (\mu)\) and \(F\) ensure that \(S\) is narrow? They show, in general, the domination problem in narrow operators has a negative answer (Example 3.3). They obtain a positive result when the norm on \(F\) is order continuous. Theorem: Let \((\Omega, \Sigma, \mu)\) be a probability space with an atomless measure \(\mu\) and \(E (\mu)\) a Köthe function space on \((\Omega, \Sigma, \mu)\). Let \(F\) be an order continuous Banach lattice and \(0 \leq S \leq T:\) \(E \to F\) be two positive operators. If \(T\) is narrow, then \(S\) is narrow.
scientific article; zbMATH DE number 2021995