If $T:X\to Y$ is continuous and $T^{t}:Y^{*}\to X^{*}$ is compact, is it true that $T$ is compact?

If $T:X\to Y$ is continuous and $T^{t}:Y^{*}\to X^{*}$ is compact, is it
true that $T$ is compact?

I have a question. I have Banach spaces $X$ and $Y$, and $Y$ is reflexive.
If $T:X\to Y$ is continuous, and $T^{t}:Y^{*}\to X^{*}, T^{t}(\phi)=\phi
\circ T$ is compact, is it true that $T$ is compact?
Thanks so much