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)
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Hint. $\displaystyle (1+x)(1-x+x^2-x^3+...) = 1$
I'm lost as to how to use this to find a solution for the question.
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.
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