It is my understanding that when an operator is densely defined, then the adjoint operator exists. But, if $X,\, Y$ Banach spaces and $A:D(A)\subset X\to Y$ linear operator.
Why is it necessary for $D(A)$ to be dense in $X$ to have an existence of the adjoint $A^*$?
If $D(A)$ is not dense in $X$, not necessarily the adjoint exists, right?