My question is : is the following reasoning correct ?
Let $X$ and $Y$ be two Banach spaces and let $T:X\rightarrow Y$ a bounded surjective operator. Then $T^{\ast }$ $:Y^{\ast }\rightarrow X^{\ast }$ is a bounded below operator and hence it has a left inverse $R^{\ast }$ $:Y^{\ast }\rightarrow X^{\ast }$, that is $R^{\ast }T^{\ast }=Id_{Y^{\ast }}$ then $% TR=Id_{X}$ and hence $T$ has a right inverse.
Or must $X$ and $Y$ be reflexive to do like ?
Thank you !