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Problem 5. The operator T : H → H is an isometry if ||T f||...

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Advance Math

Problem 5. The operator T : H → H is an isometry if ||T f|| =||f|| for all f ∈ H.

(a) Please, prove that if T is an isometry then (T f, T g) = (f,g) for all f, g ∈ H.

(b) Now prove that if T is an isometry then T∗T =I.

(c) Now prove that if T is surjective and isometry (and thusunitary) then T T∗ = I.

(d) Give an example of an isometry T that is not unitary.Hint: consider l2(N) and themap which takes (a1, a2, . . .) to (0,a1, a2, . . .).

(e) Now Prove that if T∗T is unitary then T is anisometry. Hint: Start with ||T f||2 =(f, T∗T f) and use Holders inequality to get ||T f|| ≤||f||. Next consider ||f|| = ||T ∗T f|| do get the oppositeinequality.

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