How does this statement follow? (adjoints on Hilbert spaces)

  • Thread starter pellman
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If A is an operator on a Hilbert space H and A* is its adjoint, then
upload_2015-8-8_5-53-38.png
. That is, the orthogonal complement of the range of A is the same subspace as the kernel of its adjoint.

Then the author I am reading says it follows that the statements "The range of A is a dense subspace of H" and "A* is injective on Dom(A*)" are equivalent. Can someone explain please?

The operators A and A* are not assumed to be bounded and so their domains may not be all of H and their domains may not be equal to each other.

It could also be that I am misreading this entirely.
 

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