If $Tcolon X to Y$ is such that $T^*colon Y^* to Y^*$, what does this imply about $T$?

Question

Let $X subset Y$ be a continuous embedding of reflexive Banach spaces (with different norms).
If $Tcolon X to Y$ is a bounded linear operator such that its adjoint $Tcolon Y^* to X^*$ satisfies
$$T^*colon Y^* to Y^*$$
does this imply that $T$ can be extended to be defined on $Y$, as a bounded linear operator? Are there any other implications of assuming the above?
I think the extension can be done due to reflexivity. But not sure what other properties we get.

s.harp · Answer

Look at $T^{**}: Y^{**}to Y^{**}$. Note that $T^{**}(iota(x))=iota(T(x))$ for all $xin X$ (here $iota: Yto Y^{**}$ is the canonical inclusion), simply because
$$T^{**}(iota(x))[f]= iota(x)[T^*(f)]=T^*(f)[x]=f(T(x))=iota (T(x)) ,[f]$$
after expanding all the relevant definitions.
Now since $Y$ is reflexive you may identify $Y$ with $Y^{**}$ (ie assume $iota$ is the identity), which yields that $T^{**}:Yto Y$ satisfies $T^{**}(x)=T(x)$ for all $xin X$, ie $T^{**}$ is an extension of $T$.

If $Tcolon X to Y$ is such that $T^colon Y^ to Y^*$, what does this imply about $T$?

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