On not open linear continuous operators between Banach spaces

Let X and Y be infinite-dimensional Banach spaces. Let $T: X → Y$ be a linear continuous operator with dense range and $T(X) ≠ Y$. It is proved that, for each $ε > 0$, there exists a quotient map $q: Y → Y1$, such that $Y1$ is an infinite-dimensional Banach space with a Schauder basis and $q ○ T$ is a nuclear operator of norm $≤ ε$. Thereby, we obtain with respect to the quotient spaces the proper analogue result of Kato concerning the existence of not trivial nuclear restrictions of not open linear continuous operators between Banach spaces. As a consequence, it is derived a result of Ostrovskii concerning Banach spaces which are completions with respect to total nonnorming subspaces