Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Hermitian adjoint
open-in-new
<h2 id="informal-definition">Informal definition</h2> <p>Consider a <a href="/facts/Linear_map/Q1PBKNVV">linear map</a> A : H 1 → H 2 {\displaystyle A:H_{1}\to H_{2}} between <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert spaces</a>. Without taking care of any details, the adjoint operator is the (in most cases uniquely defined) linear operator A ∗ : H 2 → H 1 {\displaystyle A^{*}:H_{2}\to H_{1}} fulfilling </p> ⟨ A h 1 , h 2 ⟩ H 2 = ⟨ h 1 , A ∗ h 2 ⟩ H 1 , {\displaystyle \left\langle Ah_{1},h_{2}\right\rangle _{H_{2}}=\left\langle h_{1},A^{*}h_{2}\right\rangle _{H_{1}},} <p>where ⟨ ⋅ , ⋅ ⟩ H i {\displaystyle \langle \cdot ,\cdot \rangle _{H_{i}}} is the <a href="/facts/Inner_product_space/HoyjElEy">inner product</a> in the Hilbert space H i {\displaystyle H_{i}} , which is linear in the first coordinate and <a href="/facts/Conjugate_linear/GTXzWMKc">conjugate linear</a> in the second coordinate. Note the special case where both Hilbert spaces are identical and A {\displaystyle A} is an operator on that Hilbert space. </p><p>When one trades the inner product for the <a href="/facts/Dual_system/2Y8KyofV">dual pairing</a>, one can define the adjoint, also called the <a href="/facts/Transpose_of_a_linear_map/uv0JWTHq">transpose</a>, of an operator A : E → F {\displaystyle A:E\to F} , where E , F {\displaystyle E,F} are <a href="/facts/Banach_spaces/sDEIk7EC">Banach spaces</a> with corresponding <a href="/facts/Norm_(mathematics)/xIbR4uE1">norms</a> ‖ ⋅ ‖ E , ‖ ⋅ ‖ F {\displaystyle \|\cdot \|_{E},\|\cdot \|_{F}} . Here (again not considering any technicalities), its adjoint operator is defined as A ∗ : F ∗ → E ∗ {\displaystyle A^{*}:F^{*}\to E^{*}} with </p> A ∗ f = f ∘ A : u ↦ f ( A u ) , {\displaystyle A^{*}f=f\circ A:u\mapsto f(Au),} <p>i.e., ( A ∗ f ) ( u ) = f ( A u ) {\displaystyle \left(A^{*}f\right)(u)=f(Au)} for f ∈ F ∗ , u ∈ E {\displaystyle f\in F^{*},u\in E} . </p><p>The above definition in the Hilbert space setting is really just an application of the Banach space case when one identifies a Hilbert space with its dual (via the <a href="/facts/Riesz_representation_theorem/9CjKDUII">Riesz representation theorem</a>). Then it is only natural that we can also obtain the adjoint of an operator A : H → E {\displaystyle A:H\to E} , where H {\displaystyle H} is a Hilbert space and E {\displaystyle E} is a Banach space. The dual is then defined as A ∗ : E ∗ → H {\displaystyle A^{*}:E^{*}\to H} with A ∗ f = h f {\displaystyle A^{*}f=h_{f}} such that </p> ⟨ h f , h ⟩ H = f ( A h ) . {\displaystyle \langle h_{f},h\rangle _{H}=f(Ah).} <h2 id="definition-for-unbounded-operators-between-banach-spaces">Definition for unbounded operators between Banach spaces</h2> <p>Let ( E , ‖ ⋅ ‖ E ) , ( F , ‖ ⋅ ‖ F ) {\displaystyle \left(E,\|\cdot \|_{E}\right),\left(F,\|\cdot \|_{F}\right)} be <a href="/facts/Banach_space/sDEIk7EC">Banach spaces</a>. Suppose A : D ( A ) → F {\displaystyle A:D(A)\to F} and D ( A ) ⊂ E {\displaystyle D(A)\subset E} , and suppose that A {\displaystyle A} is a (possibly unbounded) linear operator which is <a href="/facts/Densely_defined_operator/8uGARqJR">densely defined</a> (i.e., D ( A ) {\displaystyle D(A)} is dense in E {\displaystyle E} ). Then its adjoint operator A ∗ {\displaystyle A^{*}} is defined as follows. The domain is </p> D ( A ∗ ) := { g ∈ F ∗ : ∃ c ≥ 0 : for all u ∈ D ( A ) : | g ( A u ) | ≤ c ⋅ ‖ u ‖ E } . {\displaystyle D\left(A^{*}\right):=\left\{g\in F^{*}:~\exists c\geq 0:~{\mbox{ for all }}u\in D(A):~|g(Au)|\leq c\cdot \|u\|_{E}\right\}.} <p>Now for arbitrary but fixed g ∈ D ( A ∗ ) {\displaystyle g\in D(A^{*})} we set f : D ( A ) → R {\displaystyle f:D(A)\to \mathbb {R} } with f ( u ) = g ( A u ) {\displaystyle f(u)=g(Au)} . By choice of g {\displaystyle g} and definition of D ( A ∗ ) {\displaystyle D(A^{*})} , f is (uniformly) continuous on D ( A ) {\displaystyle D(A)} as | f ( u ) | = | g ( A u ) | ≤ c ⋅ ‖ u ‖ E {\displaystyle |f(u)|=|g(Au)|\leq c\cdot \|u\|_{E}} . Then by the <a href="/facts/Hahn%E2%80%93Banach_theorem/3AUHusvn">Hahn–Banach theorem</a>, or alternatively through extension by continuity, this yields an extension of f {\displaystyle f} , called f ^ {\displaystyle {\hat {f}}} , defined on all of E {\displaystyle E} . This technicality is necessary to later obtain A ∗ {\displaystyle A^{*}} as an operator D ( A ∗ ) → E ∗ {\displaystyle D\left(A^{*}\right)\to E^{*}} instead of D ( A ∗ ) → ( D ( A ) ) ∗ . {\displaystyle D\left(A^{*}\right)\to (D(A))^{*}.} Remark also that this does not mean that A {\displaystyle A} can be extended on all of E {\displaystyle E} but the extension only worked for specific elements g ∈ D ( A ∗ ) {\displaystyle g\in D\left(A^{*}\right)} . </p><p>Now, we can define the adjoint of A {\displaystyle A} as </p> A ∗ : F ∗ ⊃ D ( A ∗ ) → E ∗ g ↦ A ∗ g = f ^ . {\displaystyle {\begin{aligned}A^{*}:F^{*}\supset D(A^{*})&\to E^{*}\\g&\mapsto A^{*}g={\hat {f}}.\end{aligned}}} <p>The fundamental defining identity is thus </p> g ( A u ) = ( A ∗ g ) ( u ) {\displaystyle g(Au)=\left(A^{*}g\right)(u)} for u ∈ D ( A ) . {\displaystyle u\in D(A).} <h2 id="definition-for-bounded-operators-between-hilbert-spaces">Definition for bounded operators between Hilbert spaces</h2> <p>Suppose H is a complex <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a>, with <a href="/facts/Inner_product/HoyjElEy">inner product</a> ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } . Consider a <a href="/facts/Continuous_function_(topology)/sKbl02pB">continuous</a> <a href="/facts/Linear_operator/Q1PBKNVV">linear operator</a> <i>A</i> : <i>H</i> → <i>H</i> (for linear operators, continuity is equivalent to being a <a href="/facts/Bounded_operator/LYmDTIqR">bounded operator</a>). Then the adjoint of A is the continuous linear operator <i>A</i>∗ : <i>H</i> → <i>H</i> satisfying </p> ⟨ A x , y ⟩ = ⟨ x , A ∗ y ⟩ for all x , y ∈ H . {\displaystyle \langle Ax,y\rangle =\left\langle x,A^{*}y\right\rangle \quad {\mbox{for all }}x,y\in H.} <p>Existence and uniqueness of this operator follows from the <a href="/facts/Riesz_representation_theorem/9CjKDUII">Riesz representation theorem</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p><p>This can be seen as a generalization of the <i>adjoint</i> matrix of a square matrix which has a similar property involving the standard complex inner product. </p> <h2 id="properties">Properties</h2> <p>The following properties of the Hermitian adjoint of <a href="/facts/Bounded_operator/LYmDTIqR">bounded operators</a> are immediate:<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> <ol><li><a href="/facts/Involution_(mathematics)/kKxYrUVa">Involutivity</a>: <i>A</i>∗∗ = <i>A</i></li> <li>If A is invertible, then so is <i>A</i>∗, with ( A ∗ ) − 1 = ( A − 1 ) ∗ {\textstyle \left(A^{*}\right)^{-1}=\left(A^{-1}\right)^{*}} </li> <li><a href="/facts/Antilinear_map/GTXzWMKc">Conjugate linearity</a>: <ul><li>(<i>A</i> + <i>B</i>)∗ = <i>A</i>∗ + <i>B</i>∗</li> <li>(<i>λA</i>)∗ = <i>λ</i><i>A</i>∗, where <i>λ</i> denotes the <a href="/facts/Complex_conjugate/7TEaoQDe">complex conjugate</a> of the <a href="/facts/Complex_number/2w7mqI7e">complex number</a> <i>λ</i></li></ul></li> <li>"<a href="/facts/Distributive_property/0LNBrVJl">Anti-distributivity</a>": (<i>AB</i>)∗ = <i>B</i>∗<i>A</i>∗</li></ol> <p>If we define the <a href="/facts/Operator_norm/a5l0TkFW">operator norm</a> of A by </p> ‖ A ‖ op := sup { ‖ A x ‖ : ‖ x ‖ ≤ 1 } {\displaystyle \|A\|_{\text{op}}:=\sup \left\{\|Ax\|:\|x\|\leq 1\right\}} <p>then </p> ‖ A ∗ ‖ op = ‖ A ‖ op . {\displaystyle \left\|A^{*}\right\|_{\text{op}}=\|A\|_{\text{op}}.} <a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> <p>Moreover, </p> ‖ A ∗ A ‖ op = ‖ A ‖ op 2 . {\displaystyle \left\|A^{*}A\right\|_{\text{op}}=\|A\|_{\text{op}}^{2}.} <a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> <p>One says that a norm that satisfies this condition behaves like a "largest value", extrapolating from the case of self-adjoint operators. </p><p>The set of bounded linear operators on a complex Hilbert space H together with the adjoint operation and the operator norm form the prototype of a <a href="/facts/C*-algebra/B3tnGHIo">C*-algebra</a>. </p> <h2 id="adjoint-of-densely-defined-unbounded-operators-between-hilbert-spaces">Adjoint of densely defined unbounded operators between Hilbert spaces</h2> <h3>Definition</h3> <p>Let the inner product ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } be linear in the <i>first</i> argument. A <a href="/facts/Densely_defined_operator/8uGARqJR">densely defined operator</a> A from a complex Hilbert space H to itself is a linear operator whose domain <i>D</i>(<i>A</i>) is a dense <a href="/facts/Linear_subspace/2wtKTgHL">linear subspace</a> of H and whose values lie in H.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> By definition, the domain <i>D</i>(<i>A</i>∗) of its adjoint <i>A</i>∗ is the set of all <i>y</i> ∈ <i>H</i> for which there is a <i>z</i> ∈ <i>H</i> satisfying </p> ⟨ A x , y ⟩ = ⟨ x , z ⟩ for all x ∈ D ( A ) . {\displaystyle \langle Ax,y\rangle =\langle x,z\rangle \quad {\mbox{for all }}x\in D(A).} <p>Owing to the density of D ( A ) {\displaystyle D(A)} and <a href="/facts/Riesz_representation_theorem/9CjKDUII">Riesz representation theorem</a>, z {\displaystyle z} is uniquely defined, and, by definition, A ∗ y = z . {\displaystyle A^{*}y=z.} <a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p><p>Properties 1.–5. hold with appropriate clauses about <a href="/facts/Domain_of_a_function/FQvoeUpX">domains</a> and <a href="/facts/Codomain/MGc43nwQ">codomains</a>. For instance, the last property now states that (<i>AB</i>)∗ is an extension of <i>B</i>∗<i>A</i>∗ if A, B and AB are densely defined operators.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p> <h3>ker A*=(im A)⊥</h3> <p>For every y ∈ ker A ∗ , {\displaystyle y\in \ker A^{*},} the linear functional x ↦ ⟨ A x , y ⟩ = ⟨ x , A ∗ y ⟩ {\displaystyle x\mapsto \langle Ax,y\rangle =\langle x,A^{*}y\rangle } is identically zero, and hence y ∈ ( im A ) ⊥ . {\displaystyle y\in (\operatorname {im} A)^{\perp }.} </p><p>Conversely, the assumption that y ∈ ( im A ) ⊥ {\displaystyle y\in (\operatorname {im} A)^{\perp }} causes the functional x ↦ ⟨ A x , y ⟩ {\displaystyle x\mapsto \langle Ax,y\rangle } to be identically zero. Since the functional is obviously bounded, the definition of A ∗ {\displaystyle A^{*}} assures that y ∈ D ( A ∗ ) . {\displaystyle y\in D(A^{*}).} The fact that, for every x ∈ D ( A ) , {\displaystyle x\in D(A),} ⟨ A x , y ⟩ = ⟨ x , A ∗ y ⟩ = 0 {\displaystyle \langle Ax,y\rangle =\langle x,A^{*}y\rangle =0} shows that A ∗ y ∈ D ( A ) ⊥ = D ( A ) ¯ ⊥ = { 0 } , {\displaystyle A^{*}y\in D(A)^{\perp }={\overline {D(A)}}^{\perp }=\{0\},} given that D ( A ) {\displaystyle D(A)} is dense. </p><p>This property shows that ker A ∗ {\displaystyle \operatorname {ker} A^{*}} is a topologically closed subspace even when D ( A ∗ ) {\displaystyle D(A^{*})} is not. </p> <h3>Geometric interpretation</h3> <p>If H 1 {\displaystyle H_{1}} and H 2 {\displaystyle H_{2}} are Hilbert spaces, then H 1 ⊕ H 2 {\displaystyle H_{1}\oplus H_{2}} is a Hilbert space with the inner product </p> ⟨ ( a , b ) , ( c , d ) ⟩ H 1 ⊕ H 2 = def ⟨ a , c ⟩ H 1 + ⟨ b , d ⟩ H 2 , {\displaystyle {\bigl \langle }(a,b),(c,d){\bigr \rangle }_{H_{1}\oplus H_{2}}{\stackrel {\text{def}}{=}}\langle a,c\rangle _{H_{1}}+\langle b,d\rangle _{H_{2}},} <p>where a , c ∈ H 1 {\displaystyle a,c\in H_{1}} and b , d ∈ H 2 . {\displaystyle b,d\in H_{2}.} </p><p>Let J : H ⊕ H → H ⊕ H {\displaystyle J\colon H\oplus H\to H\oplus H} be the <a href="/facts/Symplectic_matrix/kkWvzo5M">symplectic mapping</a>, i.e. J ( ξ , η ) = ( − η , ξ ) . {\displaystyle J(\xi ,\eta )=(-\eta ,\xi ).} Then the graph </p> G ( A ∗ ) = { ( x , y ) ∣ x ∈ D ( A ∗ ) , y = A ∗ x } ⊆ H ⊕ H {\displaystyle G(A^{*})=\{(x,y)\mid x\in D(A^{*}),\ y=A^{*}x\}\subseteq H\oplus H} <p>of A ∗ {\displaystyle A^{*}} is the <a href="/facts/Orthogonal_complement/a13oQBsz">orthogonal complement</a> of J G ( A ) : {\displaystyle JG(A):} </p> G ( A ∗ ) = ( J G ( A ) ) ⊥ = { ( x , y ) ∈ H ⊕ H : ⟨ ( x , y ) , ( − A ξ , ξ ) ⟩ H ⊕ H = 0 ∀ ξ ∈ D ( A ) } . {\displaystyle G(A^{*})=(JG(A))^{\perp }=\{(x,y)\in H\oplus H:{\bigl \langle }(x,y),(-A\xi ,\xi ){\bigr \rangle }_{H\oplus H}=0\;\;\forall \xi \in D(A)\}.} <p>The assertion follows from the equivalences </p> ⟨ ( x , y ) , ( − A ξ , ξ ) ⟩ = 0 ⇔ ⟨ A ξ , x ⟩ = ⟨ ξ , y ⟩ , {\displaystyle {\bigl \langle }(x,y),(-A\xi ,\xi ){\bigr \rangle }=0\quad \Leftrightarrow \quad \langle A\xi ,x\rangle =\langle \xi ,y\rangle ,} <p>and </p> [ ∀ ξ ∈ D ( A ) ⟨ A ξ , x ⟩ = ⟨ ξ , y ⟩ ] ⇔ x ∈ D ( A ∗ ) & y = A ∗ x . {\displaystyle {\Bigl [}\forall \xi \in D(A)\ \ \langle A\xi ,x\rangle =\langle \xi ,y\rangle {\Bigr ]}\quad \Leftrightarrow \quad x\in D(A^{*})\ \&\ y=A^{*}x.} <h4>Corollaries</h4> <h5>A* is closed</h5> <p>An operator A {\displaystyle A} is <i>closed</i> if the graph G ( A ) {\displaystyle G(A)} is topologically closed in H ⊕ H . {\displaystyle H\oplus H.} The graph G ( A ∗ ) {\displaystyle G(A^{*})} of the adjoint operator A ∗ {\displaystyle A^{*}} is the orthogonal complement of a subspace, and therefore is closed. </p> <h5>A* is densely defined ⇔ A is closable</h5> <p>An operator A {\displaystyle A} is <i>closable</i> if the topological closure G cl ( A ) ⊆ H ⊕ H {\displaystyle G^{\text{cl}}(A)\subseteq H\oplus H} of the graph G ( A ) {\displaystyle G(A)} is the graph of a function. Since G cl ( A ) {\displaystyle G^{\text{cl}}(A)} is a (closed) linear subspace, the word "function" may be replaced with "linear operator". For the same reason, A {\displaystyle A} is closable if and only if ( 0 , v ) ∉ G cl ( A ) {\displaystyle (0,v)\notin G^{\text{cl}}(A)} unless v = 0. {\displaystyle v=0.} </p><p>The adjoint A ∗ {\displaystyle A^{*}} is densely defined if and only if A {\displaystyle A} is closable. This follows from the fact that, for every v ∈ H , {\displaystyle v\in H,} </p> v ∈ D ( A ∗ ) ⊥ ⇔ ( 0 , v ) ∈ G cl ( A ) , {\displaystyle v\in D(A^{*})^{\perp }\ \Leftrightarrow \ (0,v)\in G^{\text{cl}}(A),} <p>which, in turn, is proven through the following chain of equivalencies: </p> v ∈ D ( A ∗ ) ⊥ ⟺ ( v , 0 ) ∈ G ( A ∗ ) ⊥ ⟺ ( v , 0 ) ∈ ( J G ( A ) ) cl = J G cl ( A ) ⟺ ( 0 , − v ) = J − 1 ( v , 0 ) ∈ G cl ( A ) ⟺ ( 0 , v ) ∈ G cl ( A ) . {\displaystyle {\begin{aligned}v\in D(A^{*})^{\perp }&\Longleftrightarrow (v,0)\in G(A^{*})^{\perp }\Longleftrightarrow (v,0)\in (JG(A))^{\text{cl}}=JG^{\text{cl}}(A)\\&\Longleftrightarrow (0,-v)=J^{-1}(v,0)\in G^{\text{cl}}(A)\\&\Longleftrightarrow (0,v)\in G^{\text{cl}}(A).\end{aligned}}} <h5>A** = Acl</h5> <p>The <i>closure</i> A cl {\displaystyle A^{\text{cl}}} of an operator A {\displaystyle A} is the operator whose graph is G cl ( A ) {\displaystyle G^{\text{cl}}(A)} if this graph represents a function. As above, the word "function" may be replaced with "operator". Furthermore, A ∗ ∗ = A cl , {\displaystyle A^{**}=A^{\text{cl}},} meaning that G ( A ∗ ∗ ) = G cl ( A ) . {\displaystyle G(A^{**})=G^{\text{cl}}(A).} </p><p>To prove this, observe that J ∗ = − J , {\displaystyle J^{*}=-J,} i.e. ⟨ J x , y ⟩ H ⊕ H = − ⟨ x , J y ⟩ H ⊕ H , {\displaystyle \langle Jx,y\rangle _{H\oplus H}=-\langle x,Jy\rangle _{H\oplus H},} for every x , y ∈ H ⊕ H . {\displaystyle x,y\in H\oplus H.} Indeed, </p> ⟨ J ( x 1 , x 2 ) , ( y 1 , y 2 ) ⟩ H ⊕ H = ⟨ ( − x 2 , x 1 ) , ( y 1 , y 2 ) ⟩ H ⊕ H = ⟨ − x 2 , y 1 ⟩ H + ⟨ x 1 , y 2 ⟩ H = ⟨ x 1 , y 2 ⟩ H + ⟨ x 2 , − y 1 ⟩ H = ⟨ ( x 1 , x 2 ) , − J ( y 1 , y 2 ) ⟩ H ⊕ H . {\displaystyle {\begin{aligned}\langle J(x_{1},x_{2}),(y_{1},y_{2})\rangle _{H\oplus H}&=\langle (-x_{2},x_{1}),(y_{1},y_{2})\rangle _{H\oplus H}=\langle -x_{2},y_{1}\rangle _{H}+\langle x_{1},y_{2}\rangle _{H}\\&=\langle x_{1},y_{2}\rangle _{H}+\langle x_{2},-y_{1}\rangle _{H}=\langle (x_{1},x_{2}),-J(y_{1},y_{2})\rangle _{H\oplus H}.\end{aligned}}} <p>In particular, for every y ∈ H ⊕ H {\displaystyle y\in H\oplus H} and every subspace V ⊆ H ⊕ H , {\displaystyle V\subseteq H\oplus H,} y ∈ ( J V ) ⊥ {\displaystyle y\in (JV)^{\perp }} if and only if J y ∈ V ⊥ . {\displaystyle Jy\in V^{\perp }.} Thus, J [ ( J V ) ⊥ ] = V ⊥ {\displaystyle J[(JV)^{\perp }]=V^{\perp }} and [ J [ ( J V ) ⊥ ] ] ⊥ = V cl . {\displaystyle [J[(JV)^{\perp }]]^{\perp }=V^{\text{cl}}.} Substituting V = G ( A ) , {\displaystyle V=G(A),} obtain G cl ( A ) = G ( A ∗ ∗ ) . {\displaystyle G^{\text{cl}}(A)=G(A^{**}).} </p> <h5>A* = (Acl)*</h5> <p>For a closable operator A , {\displaystyle A,} A ∗ = ( A cl ) ∗ , {\displaystyle A^{*}=\left(A^{\text{cl}}\right)^{*},} meaning that G ( A ∗ ) = G ( ( A cl ) ∗ ) . {\displaystyle G(A^{*})=G\left(\left(A^{\text{cl}}\right)^{*}\right).} Indeed, </p> G ( ( A cl ) ∗ ) = ( J G cl ( A ) ) ⊥ = ( ( J G ( A ) ) cl ) ⊥ = ( J G ( A ) ) ⊥ = G ( A ∗ ) . {\displaystyle G\left(\left(A^{\text{cl}}\right)^{*}\right)=\left(JG^{\text{cl}}(A)\right)^{\perp }=\left(\left(JG(A)\right)^{\text{cl}}\right)^{\perp }=(JG(A))^{\perp }=G(A^{*}).} <h3>Counterexample where the adjoint is not densely defined</h3> <p>Let H = L 2 ( R , l ) , {\displaystyle H=L^{2}(\mathbb {R} ,l),} where l {\displaystyle l} is the linear measure. Select a measurable, bounded, non-identically zero function f ∉ L 2 , {\displaystyle f\notin L^{2},} and pick φ 0 ∈ L 2 ∖ { 0 } . {\displaystyle \varphi _{0}\in L^{2}\setminus \{0\}.} Define </p> A φ = ⟨ f , φ ⟩ φ 0 . {\displaystyle A\varphi =\langle f,\varphi \rangle \varphi _{0}.} <p>It follows that D ( A ) = { φ ∈ L 2 ∣ ⟨ f , φ ⟩ ≠ ∞ } . {\displaystyle D(A)=\{\varphi \in L^{2}\mid \langle f,\varphi \rangle \neq \infty \}.} The subspace D ( A ) {\displaystyle D(A)} contains all the L 2 {\displaystyle L^{2}} functions with compact support. Since 1 [ − n , n ] ⋅ φ → L 2 φ , {\displaystyle \mathbf {1} _{[-n,n]}\cdot \varphi \ {\stackrel {L^{2}}{\to }}\ \varphi ,} A {\displaystyle A} is densely defined. For every φ ∈ D ( A ) {\displaystyle \varphi \in D(A)} and ψ ∈ D ( A ∗ ) , {\displaystyle \psi \in D(A^{*}),} </p> ⟨ φ , A ∗ ψ ⟩ = ⟨ A φ , ψ ⟩ = ⟨ ⟨ f , φ ⟩ φ 0 , ψ ⟩ = ⟨ f , φ ⟩ ⋅ ⟨ φ 0 , ψ ⟩ = ⟨ φ , ⟨ φ 0 , ψ ⟩ f ⟩ . {\displaystyle \langle \varphi ,A^{*}\psi \rangle =\langle A\varphi ,\psi \rangle =\langle \langle f,\varphi \rangle \varphi _{0},\psi \rangle =\langle f,\varphi \rangle \cdot \langle \varphi _{0},\psi \rangle =\langle \varphi ,\langle \varphi _{0},\psi \rangle f\rangle .} <p>Thus, A ∗ ψ = ⟨ φ 0 , ψ ⟩ f . {\displaystyle A^{*}\psi =\langle \varphi _{0},\psi \rangle f.} The definition of adjoint operator requires that Im A ∗ ⊆ H = L 2 . {\displaystyle \mathop {\text{Im}} A^{*}\subseteq H=L^{2}.} Since f ∉ L 2 , {\displaystyle f\notin L^{2},} this is only possible if ⟨ φ 0 , ψ ⟩ = 0. {\displaystyle \langle \varphi _{0},\psi \rangle =0.} For this reason, D ( A ∗ ) = { φ 0 } ⊥ . {\displaystyle D(A^{*})=\{\varphi _{0}\}^{\perp }.} Hence, A ∗ {\displaystyle A^{*}} is not densely defined and is identically zero on D ( A ∗ ) . {\displaystyle D(A^{*}).} As a result, A {\displaystyle A} is not closable and has no second adjoint A ∗ ∗ . {\displaystyle A^{**}.} </p> <h2 id="hermitian-operators">Hermitian operators</h2> <p>A <a href="/facts/Bounded_operator/LYmDTIqR">bounded operator</a> <i>A</i> : <i>H</i> → <i>H</i> is called Hermitian or <a href="/facts/Self-adjoint_operator/VrH1nEAK">self-adjoint</a> if </p> A = A ∗ {\displaystyle A=A^{*}} <p>which is equivalent to </p> ⟨ A x , y ⟩ = ⟨ x , A y ⟩ for all x , y ∈ H . {\displaystyle \langle Ax,y\rangle =\langle x,Ay\rangle {\mbox{ for all }}x,y\in H.} <a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> <p>In some sense, these operators play the role of the <a href="/facts/Real_number/R02gw5Pb">real numbers</a> (being equal to their own "complex conjugate") and form a real <a href="/facts/Vector_space/M1pxMLx2">vector space</a>. They serve as the model of real-valued <a href="/facts/Observable/fcBAsBjm">observables</a> in <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a>. See the article on <a href="/facts/Self-adjoint_operator/VrH1nEAK">self-adjoint operators</a> for a full treatment. </p> <h2 id="adjoints-of-conjugate-linear-operators">Adjoints of conjugate-linear operators</h2> <p>For a <a href="/facts/Antilinear_map/GTXzWMKc">conjugate-linear operator</a> the definition of adjoint needs to be adjusted in order to compensate for the complex conjugation. An adjoint operator of the conjugate-linear operator A on a complex Hilbert space H is an conjugate-linear operator <i>A</i>∗ : <i>H</i> → <i>H</i> with the property: </p> ⟨ A x , y ⟩ = ⟨ x , A ∗ y ⟩ ¯ for all x , y ∈ H . {\displaystyle \langle Ax,y\rangle ={\overline {\left\langle x,A^{*}y\right\rangle }}\quad {\text{for all }}x,y\in H.} <h2 id="other-adjoints">Other adjoints</h2> <p>The equation </p> ⟨ A x , y ⟩ = ⟨ x , A ∗ y ⟩ {\displaystyle \langle Ax,y\rangle =\left\langle x,A^{*}y\right\rangle } <p>is formally similar to the defining properties of pairs of <a href="/facts/Adjoint_functor/bJ1P7AN2">adjoint functors</a> in <a href="/facts/Category_theory/6zS3DXPa">category theory</a>, and this is where adjoint functors got their name. </p> <h2 id="see-also">See also</h2> <ul><li>Mathematical concepts <ul><li><a href="/facts/Conjugate_transpose/7eBWtqVG">Conjugate transpose</a></li> <li><a href="/facts/Hermitian_operator/VrH1nEAK">Hermitian operator</a></li> <li><a href="/facts/Pullback/AcMfwboF">Pullback § Functional analysis</a></li> <li><a href="/facts/Transpose/8wmsagGS">Transpose of linear maps</a></li></ul></li> <li>Physical applications <ul><li><a href="/facts/Operator_(physics)/rCJw8UFz">Operator (physics)</a></li> <li><a href="/facts/%E2%80%A0-algebra/B3tnGHIo">†-algebra</a></li></ul></li></ul> <ul><li>Brezis, Haim (2011), <i>Functional Analysis, Sobolev Spaces and Partial Differential Equations</i> (first ed.), Springer, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-70913-0.</li> <li>Reed, Michael; Simon, Barry (2003), <i>Functional Analysis</i>, Elsevier, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 981-4141-65-8.</li> <li><a href="/facts/Walter_Rudin/9mxRAECY">Rudin, Walter</a> (1991). <a href="https://archive.org/details/functionalanalys00rudi"><i>Functional Analysis</i></a>. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: <a href="/facts/McGraw-Hill_Science/Engineering/Math/srSyQSsY">McGraw-Hill Science/Engineering/Math</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-07-054236-5. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/21163277">21163277</a>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Miller, David A. B. (2008). Quantum Mechanics for Scientists and Engineers. Cambridge University Press. pp. 262, 280. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Reed & Simon 2003, pp. 186–187; Rudin 1991, §12.9 - Reed, Michael; Simon, Barry (2003), Functional Analysis, Elsevier, ISBN 981-4141-65-8 <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Reed & Simon 2003, pp. 186–187; Rudin 1991, §12.9 - Reed, Michael; Simon, Barry (2003), Functional Analysis, Elsevier, ISBN 981-4141-65-8 <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Reed & Simon 2003, pp. 186–187; Rudin 1991, §12.9 - Reed, Michael; Simon, Barry (2003), Functional Analysis, Elsevier, ISBN 981-4141-65-8 <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Reed & Simon 2003, pp. 186–187; Rudin 1991, §12.9 - Reed, Michael; Simon, Barry (2003), Functional Analysis, Elsevier, ISBN 981-4141-65-8 <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>See unbounded operator for details. <a href="/wiki/Unbounded_operator" target="_blank">/wiki/Unbounded_operator</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Reed & Simon 2003, p. 252; Rudin 1991, §13.1 - Reed, Michael; Simon, Barry (2003), Functional Analysis, Elsevier, ISBN 981-4141-65-8 <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Rudin 1991, Thm 13.2 - Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. <a href="https://archive.org/details/functionalanalys00rudi" target="_blank">https://archive.org/details/functionalanalys00rudi</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Reed & Simon 2003, pp. 187; Rudin 1991, §12.11 - Reed, Michael; Simon, Barry (2003), Functional Analysis, Elsevier, ISBN 981-4141-65-8 <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> </ol>