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| In [[mathematics]], specifically in [[functional analysis]], each bounded [[linear operator]] on a [[Hilbert space]] has a corresponding '''adjoint operator'''. Adjoints of operators generalize [[conjugate transpose]]s of [[square matrices]] to (possibly) infinite-dimensional situations. If one thinks of operators on a Hilbert space as "generalized complex numbers", then the adjoint of an operator plays the role of the [[complex conjugate]] of a complex number.
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| The adjoint of an operator {{mvar|A}} is also sometimes called the '''Hermitian conjugate''' (after [[Charles Hermite]]) of {{mvar|A}} and is denoted by {{math|''A''*}} or {{math|''A''<sup>†</sup>}} (the latter especially when used in conjunction with the [[bra–ket notation]]).
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| ==Definition for bounded operators==<!-- This section is linked from [[Dipole]] -->
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| Suppose {{mvar|H}} is a [[Hilbert space]], with [[inner product]] <math>\langle\cdot,\cdot\rangle</math>. Consider a [[continuous function (topology)|continuous]] [[linear operator]] {{math|''A'' : ''H'' → ''H''}} (for linear operators, continuity is equivalent to being a [[bounded operator]]). Then the adjoint of {{mvar|A}} is the continuous linear operator {{math|''A''* : ''H'' → ''H''}} satisfying
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| : <math> \langle Ax , y \rangle = \langle x , A^* y \rangle \quad \mbox{for all } x,y\in H.</math>
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| Existence and uniqueness of this operator follows from the [[Riesz representation theorem]].<ref name=rs186>{{harvnb|Reed|Simon|2003|pp=186–187}}; {{harvnb|Rudin|1991|loc=§12.9}}</ref>
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| This can be seen as a generalization of the ''adjoint'' matrix of a square matrix which has a similar property involving the standard complex inner product.
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| ==Properties==
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| The following properties of the Hermitian adjoint of [[bounded operator]]s are immediate:<ref name=rs186 />
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| # {{math|1=''A''** = ''A''}} – [[involution (mathematics)|involutiveness]]
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| # If {{mvar|A}} is invertible, then so is {{math|''A''*}}, with {{math|1=(''A''*)<sup>−1</sup> = (''A''<sup>−1</sup>)*}}
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| # {{math|1=(''A'' + ''B'')* = ''A''* + ''B''*}}
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| # {{math|1=(λ''A'')* = {{conjugate|λ}}''A''*}}, where {{conjugate|λ}} denotes the [[complex conjugate]] of the [[complex number]] {{math|λ}} – [[antilinear map|antilinearity]] (together with 3.)
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| # {{math|1=(''AB'')* = ''B''* ''A''*}}
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| If we define the [[operator norm]] of {{mvar|A}} by
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| :<math> \| A \| _{op} := \sup \{ \|Ax \| : \| x \| \le 1 \} </math> | |
| then
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| :<math> \| A^* \| _{op} = \| A \| _{op}. </math><ref name=rs186 />
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| Moreover,
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| :<math> \| A^* A \| _{op} = \| A \| _{op}^2. </math><ref name=rs186 />
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| One says that a norm that satisfies this condition behaves like a "largest value", extrapolating from the case of self-adjoint operators.
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| The set of bounded linear operators on a Hilbert space {{mvar|H}} together with the adjoint operation and the operator norm form the prototype of a [[C*-algebra]].
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| == Adjoint of densely defined operators ==
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| A [[densely defined operator]] {{mvar|A}} on a Hilbert space {{mvar|H}} is a linear operator whose domain {{math|D(A)}} is a dense [[linear subspace]] of {{mvar|H}} and whose co-domain is {{mvar|H}}.<ref>See [[unbounded operator]] for details.</ref> Its adjoint {{math|''A''*}} has as domain {{math|''D''(''A''*)}} the set of all {{math|''y'' ∈ ''H''}} for which there is a {{math|''z'' ∈ ''H''}} satisfying
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| : <math> \langle Ax , y \rangle = \langle x , z \rangle \quad \mbox{for all } x \in H,</math>
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| and {{math|''A''*(''y'')}} equals the {{mvar|z}} defined thus.<ref>{{harvnb|Reed|Simon|2003|pp=252}}; {{harvnb|Rudin|1991|loc=§13.1}}</ref> | |
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| Properties 1.–5. hold with appropriate clauses about [[domain of a function|domains]] and [[codomain]]s. For instance, the last property now states that {{math|(''AB'')*}} is an extension of {{math|''B''*''A''*}} if {{mvar|A}}, {{mvar|B}} and {{mvar|AB}} are densely defined operators.<ref>{{harvnb|Rudin|1991|loc=Thm 13.2}}</ref>
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| The relationship between the image of {{mvar|A}} and the [[Kernel (linear operator)|kernel]] of its adjoint is given by: | |
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| :<math> \ker A^* = \left( \operatorname{im}\ A \right)^\bot</math> (see [[orthogonal complement]])
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| :<math> \left( \ker A^* \right)^\bot = \overline{\operatorname{im}\ A}</math>
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| Proof of the first equation:<ref>See {{harvnb|Rudin|1991|loc=Thm 12.10}} for the case of bounded operators</ref>
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| :<math>\begin{align}
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| A^* x = 0 &\iff
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| \langle A^*x,y \rangle = 0 \quad \forall y \in H \\ &\iff
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| \langle x,Ay \rangle = 0 \quad \forall y \in H \\ &\iff
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| x\ \bot \ \operatorname{im}\ A
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| \end{align}</math>
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| The second equation follows from the first by taking the orthogonal complement on both sides. Note that in general, the image need not be closed, but the kernel of a [[continuous function|continuous]] operator<ref>The same as a bounded operator.</ref> always is.
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| ==Hermitian operators==
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| A [[bounded operator]] {{math|''A'' : ''H'' → ''H''}} is called Hermitian or [[self-adjoint]] if
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| :<math> A = A^{*} </math>
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| which is equivalent to
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| :<math> \lang Ax , y \rang = \lang x , A y \rang \mbox{ for all } x,y\in H. </math><ref>{{harvnb|Reed|Simon|2003|pp=187}}; {{harvnb|Rudin|1991|loc=§12.11}}</ref>
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| In some sense, these operators play the role of the [[real number]]s (being equal to their own "complex conjugate") and form a real [[vector space]]. They serve as the model of real-valued [[observable]]s in [[quantum mechanics]]. See the article on [[self-adjoint operator]]s for a full treatment.
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| ==Adjoints of antilinear operators==
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| For an [[Antilinear map|antilinear operator]] the definition of adjoint needs to be adjusted in order to compensate for the complex conjugation. An adjoint operator of the antilinear operator {{mvar|A}} on a Hilbert space {{mvar|H}} is an antilinear operator {{math|''A''* : ''H'' → ''H''}} with the property:
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| : <math> \lang Ax , y \rang = \overline{\lang x , A^* y \rang} \quad \text{for all } x,y\in H.</math>
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| ==Other adjoints==
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| The equation
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| : <math> \lang Ax , y \rang = \lang x , A^* y \rang </math>
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| is formally similar to the defining properties of pairs of [[adjoint functor]]s in [[category theory]], and this is where adjoint functors got their name from.
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| ==See also==
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| * Mathematical concepts
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| ** [[Inner product]]
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| ** [[Hilbert space]]
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| ** [[Hermitian operator]]
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| ** [[Norm (mathematics)]]
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| ** [[Operator norm]]
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| ** [[Transpose#Transpose of linear maps|Transpose of linear maps]]
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| * Physical applications
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| ** [[Bra–ket notation]]
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| ** [[Operator (physics)]]
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| ** [[Observable]]
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| ==Footnotes==
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| {{reflist}}
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| ==References==
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| * {{citation | first1 = Michael | last1 = Reed | first2 = Barry | last2 = Simon | title = Functional Analysis | publisher = Elsevier | year = 2003 | isbn = 981-4141-65-8 }}.
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| * {{citation | first1 = Walter | last1 = Rudin | title = Functional Analysis | publisher = McGraw-Hill | year = 1991 | edition = second | isbn = 0-07-054236-8 }}.
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| {{Functional Analysis}}
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| {{DEFAULTSORT:Hermitian Adjoint}}
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| [[Category:Operator theory]]
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