U + Id)/i and T = i(U + Id) ?1 ? i. By theorem 5.3.1, there exists a measured space (X, µ), a function g: X ?? C, |g| = 1 measurable and an isomorphism ?: L 2 (X, µ) ?? H such that U = ?A g ? * . Since U ? Id is injective, A g ?1 is injective, and so the set {x ? X; g(x) = 1} is µ-negligeable, Then (U ? Id) ?1 = (?A g ?1 ? * ) ?1 = (? * ) ?1 A g ? ?1 ?A g ? * = U . Thus T = ?A f ? * where ,