# Math Help - Inverse of Mapping from Hilbert Space to Hilbert Space exists

1. ## 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)

2. Hint.

$(1+x)(1-x+x^2-x^3+...) = 1$

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

4. Let $K=T^*\circ T, \ K^2=K\circ K,
K^3=K\circ K\circ K$
etc

and consider

$I-K+K^2-K^3+...$

You 'll also need to guarantee this converges.