Let $X \subset Y$ be a continuous embedding of reflexive Banach spaces (with different norms).
If $T\colon X \to Y$ is a bounded linear operator such that its adjoint $T\colon Y^* \to X^*$ 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.
Look at $T^{**}: Y^{**}\to Y^{**}$. Note that $T^{**}(\iota(x))=\iota(T(x))$ for all $x\in X$ (here $\iota: Y\to Y^{**}$ 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 $T^{**}:Y\to Y$ satisfies $T^{**}(x)=T(x)$ for all $x\in X$, ie $T^{**}$ is an extension of $T$.