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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 !

Djalal Ounadjela
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    If an operator is bounded below, then it doesn't necessarily have a left inverse. This is where your argument breaks down. – Norbert Jul 21 '17 at 17:40

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