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

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 thus unitary) then T T^∗ = I.

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

(e) Now Prove that if T^∗T is unitary then T is an isometry. 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 opposite inequality.

Expert Solution

Colby Messinger answered 2 years ago

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