Wide operators on Köthe function spaces (Q2354538)

Let \(E\) be a Köthe function space on an atomless finite measure space \((\Omega, \Sigma, \mu)\). A function \(x \in E\) is called a \textit{mean zero sign on \(A \in \Sigma\)} if \(x\) is equal to zero outside of \(A\), equals \(\pm 1\) on \(A\) and \(\mu(\{t \in A: x(t) = 1\}) = \mu(\{t \in A: x(t) = - 1\})\). A bounded linear operator \(T: E \to X\) is said to be \textit{narrow} (some authors, including the reviewer, use the name ``PP-narrow'', i.e., narrow in the sense of Plichko and Popov) if, for every \(\varepsilon > 0\) and every \(A \in \Sigma\), there is a mean zero sign \(x\) on \(A\) such that \(\|Tx\| < \varepsilon\). For a detailed exposition of the theory of narrow operators, we refer to \textit{M. Popov} and \textit{B. Randrianantoanina} [Narrow operators on function spaces and vector lattices. Berlin: de Gruyter (2013; Zbl 1258.47002)]. Motivated by the above definition, the authors introduce a new concept of \textit{wide operator} which, in contrast to the narrow ones, maps many signs to something ``big''. Namely, an operator \(T: E \to X\) is said to be wide if there is a \(\delta > 0\) such that, for every \(A \in \Sigma\), there is a mean zero sign \(x\) on \(A\) such that \(\|Tx\| \geq \delta \|x\|\). An operator \(T: E \to X\) is said to be \textit{somewhere wide} if there is an \(A \in \Sigma\) with \(\mu(A) > 0\) such that the restriction of \(T\) to \(E(A)\) is wide, otherwise \(T\) is said to be \textit{nowhere wide}. \(T\) is called \textit{hereditary wide} if, for every \(A \in \Sigma\) with \(\mu(A) > 0\) and every sub-\(\sigma\)-algebra \(\Sigma_1\) of \( \Sigma(A)\), the restriction of \(T\) to \(E(A, \Sigma_1)\) is wide. The authors study the relationship between all these classes of operators and the narrow operators. In particular, it is demonstrated that, on every rearrangement invariant space on \([0, 1]\) with an unconditional basis, there is an operator that is simultaneously narrow and wide. Simultaneously narrow and wide operators exist also on \(L_1\), which fails to possess an unconditional basis. It remains an open problem whether such ``narrow-wide'' operators exist on every Köthe function space on \([0, 1]\).