If A is the one-sided shift on l2(N), then |A| = {AA}1/2 = I. So if A = U |A|, U must be A, which is not unitary. The existence of a polar decomposition is a consequence of Douglas' lemma: The operator C can be defined by C(Bh) := Ah for all h in H, extended by continuity to the closure of Ran(B), and by zero on the orthogonal complement to all of H. The lemma then follows since AA ≤ BB implies Ker(B) ⊂ Ker(A). In particular. If AA = BB, then C is a partial isometry, which is unique if Ker(B) ⊂ Ker(C). In general, for any bounded operator A, where (AA)1/2 is the unique positive square root of AA given by the usual functional calculus. So by the lemma, we have for some partial isometry U, which is unique if Ker(A) ⊂ Ker(U). Take P to be (AA)1/2 and one obtains the polar decomposition A = UP. Notice that an analogous argument can be used to show A = P'U, where P' is positive and U a partial isometry. When H is finite-dimensional, U can be extended to a unitary operator; this is not true in general (see example above). Alternatively, the polar decomposition can be shown using the operator version of singular value decomposition. By property of the continuous functional calculus, |A| is in the C*-algebra generated by A. A similar but weaker statement holds for the partial isometry: U is in the von Neumann algebra generated by A. If A is invertible, the polar part U will be in the C*-algebra as well. Unbounded operators If A is a closed, densely defined unbounded operator between complex Hilbert spaces then it still has a (unique) polar decomposition where |A| is a (possibly unbounded) non-negative self adjoint operator with the same domain as A, and U is a partial isometry vanishing on the orthogonal complement of the range Ran(|A|). The proof uses the same lemma as above, which goes through for unbounded operators in general. If Dom(AA) = Dom(BB) and AAh = BBh for all h ∈ Dom(AA), then there exists a partial isometry U such that A = UB. U is unique if Ran(B)⊥ ⊂ Ker(U). The operator A being closed and densely defined ensures that the operator AA is self-adjoint (with dense domain) and therefore allows one to define (AA)1/2. Applying the lemma gives polar