Editing Polar decomposition (section)

==Unbounded operators==
If ''A'' is a closed, densely defined [[unbounded operator]] between complex Hilbert spaces then it still has a (unique) '''polar decomposition'''
<math display="block">
 A = U |A|,
</math>
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(''A''{{sup|*}}''A'') = dom(''B{{sup|*}}B''), and ''A''{{sup|*}}''Ah'' = ''B''{{sup|*}}''Bh'' for all ''h'' ∈ dom(''A''{{sup|*}}''A''), then there exists a partial isometry ''U'' such that ''A'' = ''UB''. ''U'' is unique if ran(''B'')<sup>⊥</sup> ⊂ ker(''U''). The operator ''A'' being closed and densely defined ensures that the operator ''A''{{sup|*}}''A'' is self-adjoint (with dense domain) and therefore allows one to define (''A''{{sup|*}}''A'')<sup>1/2</sup>. Applying the lemma gives polar decomposition.

If an unbounded operator ''A'' is [[affiliated operator|affiliated]] to a von Neumann algebra '''M''', and ''A'' = ''UP'' is its polar decomposition, then ''U'' is in '''M''' and so is the spectral projection of ''P'', 1<sub>''B''</sub>(''P''), for any Borel set ''B'' in {{closed-open|0, ∞}}.