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)

- Jun 1st 2009, 09:30 AMfrater_cpInverse 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) - Jun 2nd 2009, 02:54 AMRebesques
Hint.

$\displaystyle (1+x)(1-x+x^2-x^3+...) = 1$ - Jun 2nd 2009, 03:16 AMfrater_cp
I'm lost as to how to use this to find a solution for the question.

- Jun 2nd 2009, 08:15 PMRebesques
Let $\displaystyle K=T^*\circ T, \ K^2=K\circ K,

K^3=K\circ K\circ K$ etc

and consider

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

You 'll also need to guarantee this converges.