Domain of an adjoint operator

Question

For $t\in [0,T]$, suppose $A(t)$ be an operator from Banach space $X$ to $X$, with $D(A(t))=D~$ i.e indepenedent of $t$. Now, if $A^{*}(t)$ be adjoint operator on $X^{*}$, then is it true that $D(A^{*}(t))$ will also be independent of $t$. ? I know that $$ D(A^{*}(t))=\lbrace x^{*} \in X^{*} ~ | ~ \exists y^{*} \in X^{*}, ~ <x^{*},A(t)x>=<y^{*},x> ~ \forall x \in D(A(t))=D \rbrace $$