A note on narrow operators on complex Riesz spaces

Abstract

The present paper continues investigation of narrow operators on complex vector lattices started in a recent paper by Dzhusoeva, Huang, Pliev and Sukochev. We answer their questions and strengthen one of their main results by proving that the extension T C : E C → F C T_{\Bbb {C}}\colon E_{\Bbb {C}}\to F_{\Bbb {C}} of a linear operator T : E → F T \colon E \to F acting from a Dedekind complete real Riesz space E E to a real Banach lattice F F to their complexifications is narrow if and only if T T is.