I have difficulty proving:
If $T$ is a normal operator in a Hilbert space, $T$ is surjective if and only if $T^*$ surjective.
Please give me some help. Thank you.
Some strong hints:
Note that you have to assume that $T$ is a closed operator, which I hope is not an issue because it's typically not appropriate to use the term "normal operator" unless the operator is closed.