Editing Unitary operator (section)

== Definition ==
'''Definition 1.''' A ''unitary operator'' is a [[bounded linear operator]] {{math|''U'' : ''H'' → ''H''}} on a Hilbert space {{mvar|H}} that satisfies {{math|1=''U''*''U'' = ''UU''* = ''I''}}, where {{math|''U''*}} is the [[Hermitian adjoint|adjoint]] of {{mvar|U}}, and {{math|''I'' : ''H'' → ''H''}} is the [[identity (mathematics)|identity]] operator.

The weaker condition {{math|1=''U''*''U'' = ''I''}} defines an ''[[isometry]]''. The other weaker condition, {{math|1=''UU''* = ''I''}}, defines a ''coisometry''. Thus a unitary operator is a bounded linear operator that is both an isometry and a coisometry,<ref>{{harvnb|Halmos|1982|loc=Sect. 127, page 69}}</ref> or, equivalently, a [[surjective function|surjective]] isometry.<ref>{{harvnb|Conway|1990|loc=Proposition I.5.2}}</ref>

An equivalent definition is the following:

'''Definition 2.''' A ''unitary operator'' is a bounded linear operator {{math|''U'' : ''H'' → ''H''}} on a Hilbert space {{mvar|H}} for which the following hold:
*{{mvar|U}} is [[surjective function|surjective]], and
*{{mvar|U}} preserves the [[inner product]] of the Hilbert space, {{mvar|H}}. In other words, for all [[vector space|vector]]s {{mvar|x}} and {{mvar|y}} in {{mvar|H}} we have: 
*:<math>\langle Ux, Uy \rangle_H = \langle x, y \rangle_H.</math>

The notion of isomorphism in the [[Category theory|category]] of Hilbert spaces is captured if domain and range are allowed to differ in this definition. Isometries preserve [[Cauchy sequence]]s; hence the [[complete metric space|completeness]] property of Hilbert spaces is preserved<ref>{{harvnb|Conway|1990|loc=Definition I.5.1}}</ref>

The following, seemingly weaker, definition is also equivalent:

'''Definition 3.''' A ''unitary operator'' is a bounded linear operator {{math|''U'' : ''H'' → ''H''}} on a Hilbert space {{mvar|H}} for which the following hold:
*the range of {{mvar|U}} is [[dense set|dense]] in {{mvar|H}}, and
*{{mvar|U}} preserves the inner product of the Hilbert space, {{mvar|H}}. In other words, for all vectors {{mvar|x}} and {{mvar|y}} in {{mvar|H}} we have: 
*:<math>\langle Ux, Uy \rangle_H = \langle x, y \rangle_H.</math>

To see that definitions 1 and 3 are equivalent, notice that {{mvar|U}} preserving the inner product implies {{mvar|U}} is an [[isometry]] (thus, a [[bounded linear operator]]). The fact that {{mvar|U}} has dense range ensures it has a bounded inverse {{math|''U''<sup>−1</sup>}}. It is clear that {{math|1=''U''<sup>−1</sup> = ''U''*}}.

Thus, unitary operators are just [[automorphism]]s of Hilbert spaces, i.e., they preserve the structure (the vector space structure, the inner product, and hence the [[topology]]) of the space on which they act. The [[group (mathematics)|group]] of all unitary operators from a given Hilbert space {{mvar|H}} to itself is sometimes referred to as the ''Hilbert group'' of {{mvar|H}}, denoted {{math|Hilb(''H'')}} or {{math|''U''(''H'')}}.