Editing Unitary operator (section)

==Properties==
* The [[spectrum (functional analysis)|spectrum]] of a unitary operator {{mvar|U}} lies on the [[unit circle]]. That is, for any complex number {{mvar|λ}} in the spectrum, one has {{math|{{!}}''λ''{{!}} {{=}} 1}}. This can be seen as a consequence of the [[spectral theorem]] for [[normal operator]]s. By the theorem, {{mvar|U}} is unitarily equivalent to multiplication by a [[Borel measurable|Borel-measurable]] {{mvar|f}} on {{math|''L''<sup>2</sup>(''μ'')}}, for some finite [[measure space]] {{math|(''X'', ''μ'')}}. Now {{math|''UU''* {{=}} ''I''}} implies {{math|{{!}}''f''(''x''){{!}}<sup>2</sup> {{=}} 1}}, {{mvar|μ}}-a.e. This shows that the essential range of {{mvar|f}}, therefore the spectrum of {{mvar|U}}, lies on the unit circle.
* A linear map is unitary if it is surjective and isometric. (Use [[Polarization identity]] to show the only if part.)