Jiménez Sevilla, María Del MarLajara, SebastiánRuiz Risueño, Miguel Ángel2026-03-262026-03-262026Jiménez-Sevilla, Mar; Lajara, Sebastián; Ruiz-Risueño, Miguel Ángel; Operators with the Kato Property on Banach Spaces. Results Math. 81 (2026), no. 2, Paper No. 55.10.1007/s00025-026-02614-7https://hdl.handle.net/20.500.14352/1343722026 Acuerdos transformativos CRUEWe consider a class of bounded linear operators between Banach spaces, which we call operators with the Kato property, that includes the family of strictly singular operators between those spaces. We show that if T : E → F is a dense-range operator with that property and E has a separable quotient, then for each proper dense operator range R ⊂ E there exists a closed subspace X ⊂ E such that E/X is separable, T(X) is dense in F and R+ X is infinite-codimensional. If E∗ is weak∗-separable, the subspace X can be built so that, in addition to the former properties, R ∩ X = {0}. Some applications to the geometry of Banach spaces are given. In particular, we provide the next extensions of well-known results of Johnson and Plichko: if X and Y are quasicomplemented but not complemented subspaces of a Banach space E and X has a separable quotient, then X contains a closed subspace X1 such that dim(X/X1) = ∞ and X1 is a quasicomplement of Y , and if T : E → F is an operator with non-closed range and E has a separable quotient, then there exists a weak∗-closed subspace Z ⊂ E∗ such that T ∗(F∗) ∩ Z = {0}. Some refinements of these results, in the case that E∗ is weak∗-separable, are also given. Finally, we show that if E is a Banach space with a separable quotient, then E∗ is weak∗-separable if, and only if, for every closed subspace X ⊂ E and every proper dense operator range R ⊂ E containing X there exists a quasicomplement Y of X in E such that Y ∩ R = {0}.engAttribution 4.0 Internationalhttp://creativecommons.org/licenses/by/4.0/journal articlehttps://doi.org/10.1007/s00025-026-02614-7https://link.springer.com/article/10.1007/s00025-026-02614-7open accessFunctional AnalysisBanach spaceOperator with the Kato propertySeparable quotientProper dense operator rangeQuasicomplemented subspaceMatemáticas (Matemáticas)1202 Análisis y Análisis Funcional