Let $X$ and $Y$ be infinite-dimensional Banach spaces. Let $T:X\to Y$ be a linear continuous operator with dense range and $T(X)\not =Y$. It is proved that, for each $\epsilon>0$, there exists a quotient map $q: Y\to Y_1$, such that $Y_1$ is an infinite-dimensional Banach space with a Schauder basis and $q\circ T$ is a nuclear operator of norm $\leq \epsilon$. Thereby, we obtain with respect to quotient spaces the proper analogue result of KATO concernig 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 repsect to total nonnorming subspaces.
On not open linear continuous maps between Banach spaces
ALBANESE, Angela Anna
2006-01-01
Abstract
Let $X$ and $Y$ be infinite-dimensional Banach spaces. Let $T:X\to Y$ be a linear continuous operator with dense range and $T(X)\not =Y$. It is proved that, for each $\epsilon>0$, there exists a quotient map $q: Y\to Y_1$, such that $Y_1$ is an infinite-dimensional Banach space with a Schauder basis and $q\circ T$ is a nuclear operator of norm $\leq \epsilon$. Thereby, we obtain with respect to quotient spaces the proper analogue result of KATO concernig 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 repsect to total nonnorming subspaces.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
