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 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) - June 2nd 2009, 02:54 AMRebesques
Hint.

- June 2nd 2009, 03:16 AMfrater_cp
I'm lost as to how to use this to find a solution for the question.

- June 2nd 2009, 08:15 PMRebesques
Let etc

and consider

You 'll also need to guarantee this converges.