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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.

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Is $T$ assumed to be bounded? If so, $T$ is actually invertible, hence so is $T^*$. – Jonas Meyer Dec 29 '12 at 6:39
@JonasMeyer: Nice observation! – copper.hat Dec 29 '12 at 6:50

Some strong hints:

  1. For any normal $T$, show that the kernels of $T$ and $T^{\ast}$ coincide.
  2. Relate the ranges of $T$ and $T^{\ast}$ to their respective kernels (there is a standard theorem about this).

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.

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