Editing Operator theory (section)

===Polar decomposition===
{{Main article|Polar decomposition}}

The '''polar decomposition''' of any [[bounded linear operator]] ''A'' between complex [[Hilbert space]]s is a canonical factorization as the product of a [[partial isometry]] and a non-negative operator.<ref>{{citation|title=A Course in Operator Theory | series=[[Graduate Studies in Mathematics]]|first=John B. |last=Conway|publisher=American Mathematical Society|year= 2000 | isbn=0821820656}}</ref>

The polar decomposition for matrices generalizes as follows: if ''A'' is a bounded linear operator then there is a unique factorization of ''A'' as a product ''A'' = ''UP'' where ''U'' is a partial isometry, ''P'' is a non-negative self-adjoint operator and the initial space of ''U'' is the closure of the range of ''P''.

The operator ''U'' must be weakened to a partial isometry, rather than unitary, because of the following issues. If ''A'' is the [[shift operator|one-sided shift]] on ''l''{{i sup|2}}('''N'''), then |''A''| = (''A*A'')<sup>1/2</sup> = ''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]]:
{{math theorem | name = Lemma | math_statement = If ''A'', ''B'' are bounded operators on a Hilbert space ''H'', and ''A*A'' ≤ ''B*B'', then there exists a contraction ''C'' such that ''A'' = ''CB''. Furthermore, ''C'' is unique if ''Ker''(''B*'') ⊂ ''Ker''(''C'').}}

The operator ''C'' can be defined by {{math|1=''C''(''Bh'') = ''Ah''}}, extended by continuity to the closure of ''Ran''(''B''), and by zero on the orthogonal complement of {{math|Ran(''B'')}}. The operator ''C'' is well-defined since {{math|''A*A'' ≤ ''B*B''}} implies {{math|Ker(''B'') ⊂ Ker(''A'')}}. The lemma then follows.

In particular, if {{math|1=''A*A'' = ''B*B''}}, then ''C'' is a partial isometry, which is unique if {{math|Ker(''B*'') ⊂ Ker(''C'').}}
In general, for any bounded operator ''A'',
<math display="block">A^*A = (A^*A)^{\frac{1}{2}} (A^*A)^{\frac{1}{2}},</math>
where (''A*A'')<sup>1/2</sup> is the unique positive square root of ''A*A'' given by the usual [[functional calculus]]. So by the lemma, we have
<math display="block">A = U (A^*A)^{\frac{1}{2}}</math>
for some partial isometry ''U'', which is unique if Ker(''A'') ⊂ Ker(''U''). (Note {{math|1=Ker(''A'') = Ker(''A*A'') = Ker(''B'') = Ker(''B*'')}}, where {{math|1=''B'' = ''B*'' = (''A*A'')<sup>1/2</sup>}}.) Take ''P'' to be (''A*A'')<sup>1/2</sup> 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#Bounded operators on Hilbert spaces|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: the polar part ''U'' is in the [[von Neumann algebra]] generated by ''A''. If ''A'' is invertible, ''U'' will be in the [[C*-algebra]] generated by ''A'' as well.