1. Let X be a Banach space and let T be a bounded linear operator on...

 

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1. Let X be a Banach space and let T be a bounded linear operator on X. Show that if ||T|| < 1, then I - T has a bounded inverse (I-T)-¹ on X and it holds that ∞0 1 (I – T)-¹ = ΣTN, ||(I - T) -¯`' || ≤ 1 - ||T|| | N=0 Here the series o TN converges in B(X). N=0 1. Let X be a Banach space and let T be a bounded linear operator on X. Show that if ||T|| < 1, then I - T has a bounded inverse (I-T)-¹ on X and it holds that ∞0 1 (I – T)-¹ = ΣTN, ||(I - T) -¯`' || ≤ 1 - ||T|| | N=0 Here the series o TN converges in B(X). N=0

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Solution let X be T let operator IT II 1 then inverse IT tha... View the full answer