# Inverse of Mapping from Hilbert Space to Hilbert Space exists

• June 1st 2009, 09:30 AM
frater_cp
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)
• June 2nd 2009, 02:54 AM
Rebesques
Hint.

$(1+x)(1-x+x^2-x^3+...) = 1$
• June 2nd 2009, 03:16 AM
frater_cp
I'm lost as to how to use this to find a solution for the question.
• June 2nd 2009, 08:15 PM
Rebesques
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.