Jiménez Sevilla, María Del MarLajara López, Sebastián2024-05-232024-05-232024-03Jiménez-Sevilla, Mar, and Sebastián Lajara. "Operator ranges in Banach spaces with weak star separable dual." Journal of Mathematical Analysis and Applications 531.2 (2024): 127881.0022-247X10.1016/j.jmaa.2023.127881https://hdl.handle.net/20.500.14352/1043662023 Acuerdos transformativos CRUEWe provide several extensions for Banach spaces with weak⁎-separable dual of a theorem of Schevchik ensuring that for every proper dense operator range R in a separable Banach space E, there exists a one-to-one and dense-range operator such that . These results lead to several characterizations of Banach spaces with weak⁎-separable dual in terms of disjointness properties of operator ranges, which yield a refinement of a theorem of Plichko concerning the spaceability of the complementary set of a proper dense operator range, and an affirmative solution to a problem of Borwein and Tingley for the class of Banach spaces with a separable quotient and weak⁎-separable dual. We also provide an extension to these spaces of a theorem of Cross and Shevchik, which guarantees that for every proper dense operator range R in a separable Banach space E there exist two closed quasicomplementary subspaces X and Y of E such that ... Finally, we prove that some weak forms of the theorems of Shevchik and Cross and Shevchik do not hold in any nonseparable weakly Lindelöf determined Banach space.engAttribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/journal articleopen accessBanach space with weak star separable dualNuclear operatorOperator rangeQuasicomplemented subspaceSpaceabilityAnálisis funcional y teoría de operadores1202 Análisis y Análisis Funcional