Thread: Inverse of Mapping from Hilbert Space to Hilbert Space exists

Let S = I + T^{\ast}T : H \longrightarrow H, where T is linear and bounded.

Show that S^{-1} S(H) \longrightarrow H exists.

(I is the Identity operator)

Hint.

I'm lost as to how to use this to find a solution for the question.

Let etc

and consider



You 'll also need to guarantee this converges.