Operator ranges in Banach spaces with weak star separable dual

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We 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 T(E)∩R={0}. 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 R∩(X+Y)={0}. 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.