Editing Riesz representation theorem (section)

== Adjoints and transposes ==

Let <math>A : H \to Z</math> be a [[continuous linear operator]] between [[Hilbert space|Hilbert spaces]] <math>\left(H, \langle \cdot, \cdot \rangle_H\right)</math> and <math>\left(Z, \langle \cdot, \cdot \rangle_Z \right).</math> As before, let <math>\langle y \mid x \rangle_H := \langle x, y \rangle_H</math> and <math>\langle y \mid x \rangle_Z := \langle x, y \rangle_Z.</math> 

Denote by
<math display=block>\begin{alignat}{4}
\Phi_H :\;&& H &&\;\to    \;& H^* \\[0.3ex]
          && g &&\;\mapsto\;& \langle \,g \mid \cdot\, \rangle_H \\
\end{alignat}
\quad \text{ and } \quad 
\begin{alignat}{4}
\Phi_Z :\;&& Z &&\;\to    \;& Z^* \\[0.3ex]
          && y &&\;\mapsto\;& \langle \,y  \mid \cdot\, \rangle_Z \\
\end{alignat}</math>
the usual bijective antilinear isometries that satisfy:
<math display=block>\left(\Phi_H g\right) h = \langle g\mid h \rangle_H \quad \text{ for all } g, h \in H \qquad \text{ and } \qquad \left(\Phi_Z y\right) z = \langle y \mid z \rangle_Z \quad \text{ for all } y, z \in Z.</math>

=== Definition of the adjoint ===

{{Main|Hermitian adjoint|Conjugate transpose}}

For every <math>z \in Z,</math> the scalar-valued map <math>\langle z\mid A (\cdot) \rangle_Z</math><ref group=note name="ExplicitDefOfInnerProductOfTranspose" /> on <math>H</math> defined by 
<math display=block>h \mapsto \langle z\mid A h \rangle_Z = \langle A h, z \rangle_Z</math> 

is a continuous linear functional on <math>H</math> and so by the Riesz representation theorem, there exists a unique vector in <math>H,</math> denoted by <math>A^* z,</math> such that <math>\langle z \mid A (\cdot) \rangle_Z = \left\langle A^* z \mid \cdot\, \right\rangle_H,</math> or equivalently, such that 
<math display=block>\langle z \mid A h \rangle_Z = \left\langle A^* z \mid h \right\rangle_H \quad \text{ for all } h \in H.</math> 

The assignment <math>z \mapsto A^* z</math> thus induces a function <math>A^* : Z \to H</math> called the {{em|adjoint}} of <math>A : H \to Z</math> whose defining condition is 
<math display=block>\langle z \mid A h \rangle_Z = \left\langle A^* z\mid h \right\rangle_H \quad \text{ for all } h \in H  \text{ and all } z \in Z.</math> 
The adjoint <math>A^* : Z \to H</math> is necessarily a [[Continuous linear operator|continuous]] (equivalently, a [[Bounded linear operator|bounded]]) [[linear operator]]. 

If <math>H</math> is finite dimensional with the standard inner product and if <math>M</math> is the [[transformation matrix]] of <math>A</math> with respect to the standard orthonormal basis then <math>M</math>'s [[conjugate transpose]] <math>\overline{M^{\operatorname{T}}}</math> is the transformation matrix of the adjoint <math>A^*.</math> 

=== Adjoints are transposes ===

{{Main|Transpose of a linear map}}
{{See also|Transpose}}

It is also possible to define the {{em|[[Transpose of a linear map|transpose]]}} or {{em|[[algebraic adjoint]]}} of <math>A : H \to Z,</math> which is the map <math>{}^{t}A : Z^* \to H^*</math> defined by sending a continuous linear functionals <math>\psi \in Z^*</math> to 
<math display=block>{}^{t}A(\psi) := \psi \circ A,</math> 
where the [[Function composition|composition]] <math>\psi \circ A</math> is always a continuous linear functional on <math>H</math> and it satisfies <math>\|A\| = \left\|{}^t A\right\|</math> (this is true more generally, when <math>H</math> and <math>Z</math> are merely [[normed space]]s).{{sfn|Rudin|1991|pp=92-115}} 
<!-------------- START: Removed info --------------
which is equivalent to:
<math display=block>\psi(A h) = \left({}^t A(\psi)\right) h \quad \text{ for all } h \in H  \text{ and all } \psi \in Z^*.</math>
---------------- END:   Removed info -------------->
So for example, if <math>z \in Z</math> then <math>{}^{t}A</math> sends the continuous linear functional <math>\langle z \mid \cdot \rangle_Z \in Z^*</math> (defined on <math>Z</math> by <math>g \mapsto \langle z \mid g \rangle_Z</math>) to the continuous linear functional <math>\langle z \mid A(\cdot) \rangle_Z \in H^*</math> (defined on <math>H</math> by <math>h \mapsto \langle z \mid A(h) \rangle_Z</math>);<ref group=note name="ExplicitDefOfInnerProductOfTranspose" /> 
using bra-ket notation, this can be written as <math>{}^{t}A \langle z \mid ~=~ \langle z \mid A</math> where the juxtaposition of <math>\langle z \mid</math> with <math>A</math> on the right hand side denotes function composition: <math>H \xrightarrow{A} Z \xrightarrow{\langle z \mid} \Complex.</math> 

The adjoint <math>A^* : Z \to H</math> is actually just to the transpose <math>{}^{t}A : Z^* \to H^*</math>{{sfn|Rudin|1991|pp=306-312}} when the Riesz representation theorem is used to identify <math>Z</math> with <math>Z^*</math> and <math>H</math> with <math>H^*.</math> 

Explicitly, the relationship between the adjoint and transpose is:

{{NumBlk|:|<math>{}^{t}A ~\circ~ \Phi_Z ~=~ \Phi_H ~\circ~ A^*</math>|{{EquationRef|Adjoint-transpose}}|LnSty=1px dashed black}}

which can be rewritten as:
<math display=block>A^* ~=~ \Phi_H^{-1} ~\circ~ {}^{t}A ~\circ~ \Phi_Z \quad \text{ and } \quad {}^{t}A ~=~ \Phi_H ~\circ~ A^* ~\circ~ \Phi_Z^{-1}.</math>

{{Math proof|title=Proof|drop=hidden|proof=
To show that <math>{}^{t}A ~\circ~ \Phi_Z ~=~ \Phi_H ~\circ~ A^*,</math> fix <math>z \in Z.</math> 
The definition of <math>{}^{t}A</math> implies <math display=block>\left({}^{t}A \circ \Phi_Z\right) z = {}^{t}A \left(\Phi_Z z\right) = \left(\Phi_Z z\right) \circ A</math> so it remains to show that <math>\left(\Phi_Z z\right) \circ A = \Phi_H\left(A^* z\right).</math> If <math>h \in H</math> then <math display=block>\left(\left(\Phi_Z z\right) \circ A\right) h = \left(\Phi_Z z\right)(A h) = \langle z\mid A h \rangle_Z = \langle A^* z\mid h \rangle_H = \left(\Phi_H(A^* z)\right) h,</math> as desired. <math>\blacksquare</math>
}}

Alternatively, the value of the left and right hand sides of ({{EquationNote|Adjoint-transpose}}) at any given <math>z \in Z</math> can be rewritten in terms of the inner products as: 
<math display=block>\left({}^{t}A ~\circ~ \Phi_Z\right) z = \langle z \mid A (\cdot) \rangle_Z \quad \text{ and } \quad\left(\Phi_H ~\circ~ A^*\right) z = \langle A^* z\mid\cdot\, \rangle_H</math>
so that <math>{}^{t}A ~\circ~ \Phi_Z ~=~ \Phi_H ~\circ~ A^*</math> holds if and only if <math>\langle z \mid A (\cdot) \rangle_Z = \langle A^* z\mid\cdot\, \rangle_H</math> holds; but the equality on the right holds by definition of <math>A^* z.</math> 
The defining condition of <math>A^* z</math> can also be written
<math display=block>\langle z \mid A ~=~ \langle A^*z \mid</math> 
if bra-ket notation is used.

===Descriptions of self-adjoint, normal, and unitary operators===

Assume <math>Z = H</math> and let <math>\Phi := \Phi_H = \Phi_Z.</math> 
Let <math>A : H \to H</math> be a continuous (that is, bounded) linear operator. 

Whether or not <math>A : H \to H</math> is [[Self-adjoint operator|self-adjoint]], [[Normal operator|normal]], or [[Unitary operator|unitary]] depends entirely on whether or not <math>A</math> satisfies certain defining conditions related to its adjoint, which was shown by ({{EquationNote|Adjoint-transpose}}) to essentially be just the transpose <math>{}^t A : H^* \to H^*.</math> 
Because the transpose of <math>A</math> is a map between continuous linear functionals, these defining conditions can consequently be re-expressed entirely in terms of linear functionals, as the remainder of subsection will now describe in detail. 
The linear functionals that are involved are the simplest possible continuous linear functionals on <math>H</math> that can be defined entirely in terms of <math>A,</math> the inner product <math>\langle \,\cdot\mid\cdot\, \rangle</math> on <math>H,</math> and some given vector <math>h \in H.</math> 
Specifically, these are <math>\left\langle A h\mid\cdot\, \right\rangle</math> and <math>\langle h\mid A (\cdot) \rangle</math><ref group=note name="ExplicitDefOfInnerProductOfTranspose" /> where 
<math display=block>\left\langle A h\mid\cdot\, \right\rangle = \Phi (A h) = (\Phi \circ A) h \quad \text{ and } \quad \langle h\mid A (\cdot) \rangle = \left({}^{t}A \circ \Phi\right) h.</math> 

'''Self-adjoint operators'''

{{See also|Self-adjoint operator|Hermitian matrix|Symmetric matrix}}

A continuous linear operator <math>A : H \to H</math> is called [[Self-adjoint operator|self-adjoint]] if it is equal to its own adjoint; that is, if <math>A = A^*.</math> Using ({{EquationNote|Adjoint-transpose}}), this happens if and only if:
<math display=block>\Phi \circ A = {}^t A \circ \Phi</math>
where this equality can be rewritten in the following two equivalent forms:
<math display=block>A = \Phi^{-1} \circ {}^t A \circ \Phi \quad \text{ or } \quad {}^{t}A = \Phi \circ A \circ \Phi^{-1}.</math>

Unraveling notation and definitions produces the following characterization of self-adjoint operators in terms of the aforementioned continuous linear functionals: <math>A</math> is self-adjoint if and only if for all <math>z \in H,</math> the linear functional <math>\langle z\mid A (\cdot) \rangle</math><ref group=note name="ExplicitDefOfInnerProductOfTranspose" /> is equal to the linear functional <math>\langle A z\mid\cdot\, \rangle</math>; that is, if and only if

{{NumBlk|:|<math>\langle z\mid A (\cdot) \rangle = \langle A z\mid \cdot\, \rangle \quad \text{ for all } z \in H</math>|{{EquationRef|Self-adjointness functionals}}|LnSty=1px dashed black}}

where if bra-ket notation is used, this is
<math display=block>\langle z \mid A ~=~ \langle A z \mid \quad \text{ for all } z \in H.</math> 

'''Normal operators'''

{{See also|Normal operator|Normal matrix}}

A continuous linear operator <math>A : H \to H</math> is called [[Normal operator|normal]] if <math>A A^* = A^* A,</math> which happens if and only if for all <math>z, h \in H,</math> 
<math display=block>\left\langle A A^* z\mid h \right\rangle = \left\langle A^* A z\mid h \right\rangle.</math> 

Using ({{EquationNote|Adjoint-transpose}}) and unraveling notation and definitions produces<ref group=proof name="NormalCharFunctionals" /> the following characterization of normal operators in terms of inner products of continuous linear functionals: <math>A</math> is a normal operator if and only if  

{{NumBlk|:|<math>\left\langle \,\langle A h \mid\cdot\, \rangle\mid\langle A z \mid\cdot\, \rangle\, \right\rangle_{H^*} ~=~ \left\langle \,\langle h | A(\cdot) \rangle\mid\langle z \mid A(\cdot) \rangle\, \right\rangle_{H^*} \quad \text{ for all } z, h \in H</math>|{{EquationRef|Normality functionals}}|LnSty=1px dashed black}}
where the left hand side is also equal to <math>\overline{\langle A h \mid A z \rangle}_H = \langle A z \mid A h \rangle_H.</math> 
The left hand side of this characterization involves ''only'' linear functionals of the form <math>\langle A h \mid\cdot\, \rangle</math> while the right hand side involves ''only'' linear functions of the form <math>\langle h \mid A(\cdot) \rangle</math> (defined as above<ref group=note name="ExplicitDefOfInnerProductOfTranspose" />). 
So in plain English, characterization ({{EquationNote|Normality functionals}}) says that an operator is ''normal'' when the inner product of any two linear functions of the first form is equal to the inner product of their second form (using the same vectors <math>z, h \in H</math> for both forms).
In other words, if it happens to be the case (and when <math>A</math> is injective or self-adjoint, it is) that the assignment of linear functionals <math>\langle A h \mid\cdot\, \rangle ~\mapsto~ \langle h | A(\cdot) \rangle</math> is well-defined (or alternatively, if <math>\langle h | A(\cdot) \rangle ~\mapsto~ \langle A h \mid\cdot\, \rangle</math> is well-defined) where <math>h</math> ranges over <math>H,</math> then <math>A</math> is a normal operator if and only if this assignment preserves the inner product on <math>H^*.</math> 

The fact that every self-adjoint bounded linear operator is normal follows readily by direct substitution of <math>A^* = A</math> into either side of <math>A^* A = A A^*.</math> 
This same fact also follows immediately from the direct substitution of the equalities ({{EquationNote|Self-adjointness functionals}}) into either side of ({{EquationNote|Normality functionals}}). 

Alternatively, for a complex Hilbert space, the continuous linear operator <math>A</math> is a normal operator if and only if <math>\|Az\| = \left\|A^* z\right\|</math> for every <math>z \in H,</math>{{sfn|Rudin|1991|pp=306-312}} which happens if and only if 
<math display=block>\|Az\|_H = \|\langle z\, | \,A(\cdot) \rangle\|_{H^*} \quad \text{ for every } z \in H.</math>

'''Unitary operators'''

{{See also|Unitary transformation|Unitary matrix}}

An invertible bounded linear operator <math>A : H \to H</math> is said to be [[Unitary operator|unitary]] if its inverse is its adjoint: <math>A^{-1} = A^*.</math> 
By using ({{EquationNote|Adjoint-transpose}}), this is seen to be equivalent to <math>\Phi \circ A^{-1} = {}^{t}A \circ \Phi.</math> 
Unraveling notation and definitions, it follows that <math>A</math> is unitary if and only if  
<math display=block>\langle A^{-1} z\mid\cdot\, \rangle = \langle z\mid A (\cdot) \rangle \quad \text{ for all } z \in H.</math>

The fact that a bounded invertible linear operator <math>A : H \to H</math> is unitary if and only if <math>A^* A = \operatorname{Id}_H</math> (or equivalently, <math>{}^t A \circ \Phi \circ A = \Phi</math>) produces another (well-known) characterization: an invertible bounded linear map <math>A</math> is unitary if and only if
<math display=block>\langle A z\mid A (\cdot)\, \rangle = \langle z\mid\cdot\, \rangle \quad \text{ for all } z \in H.</math>

Because <math>A : H \to H</math> is invertible (and so in particular a bijection), this is also true of the transpose <math>{}^t A : H^* \to H^*.</math> This fact also allows the vector <math>z \in H</math> in the above characterizations to be replaced with <math>A z</math> or <math>A^{-1} z,</math> thereby producing many more equalities. Similarly, <math>\,\cdot\,</math> can be replaced with <math>A(\cdot)</math> or <math>A^{-1}(\cdot).</math>