Let $X$ and $Y$ be separable Hilbert spaces with duals $X^*$ and $Y^*$.
We have that $Y \subset X$.
Suppose $A, B \in Y^*$ and that $Ay=By$ holds for all $y \in Y$.
I think this means that $A=B$, where the equality is in $Y^*.$
Suppose now that i know that in fact $A \in X^*.$ Then is it true that $B \in X^*$?
You are saying that $ A \in X^{*}$ is such that its restriction $ A_{Y}\in Y^{*} $ is equal to $B^{*} $ , with $B^{*}\in Y^{*}$. If you are trying to extend $B^{*}$ to the whole $X^{*}$ s.t. $A^{*}=B^{*}$ on $X^{*}$, then you need to consider the Hahn-Banach theorem. First of all, you need to consider bounded operators. Moreover the extension is not unique, in general.
If $Y= \{ 0 \}$ then your condition is automatically satisfied, but there is no reasonable way of viewing $B$ as being in $X^*$, so I would say the answer is negative.
