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

[pellman]

If A is an operator on a Hilbert space H and A* is its adjoint, then

. 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.

 

[pellman]

Nevermind. It seems that what I was missing is that for a dense subspace Q of H, the orthogonal complement of Q is {0}. An explanation of this is given here http://www.math.colostate.edu/~pauld/M645/BasicHS.pdf