More on representation of operators on \(L_{1}\)
The symbol \(L_1\) is used for \(L_1[0,1]\) and \(\mathcal L(L_1)\) for the space of bounded linear operators that act from \(L_1\) to \(L_1\). An operator \(T \in \mathcal L(L_1)\) is said to be a pseudo embedding if it can be represented as a sum of a pointwise absolutely convergent series of disjointness preserving operators, and is said to be narrow in the sense of \textit{A. M. Plichko} and \textit{M. M. Popov} [Diss. Math. 306, 85 p. (1990; Zbl 0715.46011)] if, for every measurable subset \(A \subset [0, 1]\) and \(\varepsilon > 0\), there exists a decomposition of \(A\) into a disjoint union of measurable subsets \(B, C\), \(\mu(B) = \mu(C) = \frac{1}{2} \mu(A)\), in such a way that \(\|T( \mathbf 1_B - \mathbf 1_C)\| < \varepsilon\). The Kalton-Rosenthal representation theorem says that every operator \(T \in \mathcal{L}(L_1)\) has a unique representation \(T = T_{P E} + T_N\), where \(T_{P E}\) is a pseudo embedding and \(T_N\) is narrow. The main result of the paper under review gives a Daugavet-type inequality for the norm of any operator \(T \in \mathcal L(L_1)\) through the components of its Kalton-Rosenthal representation, namely, \(\|T\| \geq \|T_{P E}\|_h + \|T_N\|\), where \(\|T_{P E}\|_h\) means the infimum of norms of the restrictions of \(T_{P E}\) to all possible subspaces \(L_1(A, \Sigma_1, \mu) \subset L_1\), where \(A \subset [0, 1]\) is a measurable subset of positive measures and \(\Sigma_1\) is a \(\sigma\)-algebra of measurable subsets of \(A\) such that the restriction of the Lebesgue measure to \(\Sigma_1\) is atomless.
scientific article; zbMATH DE number 6991298