A
A*
\langleAx,y\rangle=\langlex,A*y\rangle,
where
\langle ⋅ , ⋅ \rangle
The adjoint may also be called the Hermitian conjugate or simply the Hermitian[1] after Charles Hermite. It is often denoted by in fields like physics, especially when used in conjunction with bra–ket notation in quantum mechanics. In finite dimensions where operators can be represented by matrices, the Hermitian adjoint is given by the conjugate transpose (also known as the Hermitian transpose).
The above definition of an adjoint operator extends verbatim to bounded linear operators on Hilbert spaces
H
H.
A:H1\toH2
A*:H2\toH1
\left\langleAh1,h2
\right\rangle | |
H2 |
=\left\langleh1,A*h2
\right\rangle | |
H1 |
,
where
\langle ⋅ , ⋅
\rangle | |
Hi |
Hi
A
When one trades the inner product for the dual pairing, one can define the adjoint, also called the transpose, of an operator
A:E\toF
E,F
\| ⋅ \|E,\| ⋅ \|F
A*:F*\toE*
A*f=f\circA:u\mapstof(Au),
I.e.,
\left(A*f\right)(u)=f(Au)
f\inF*,u\inE
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. Then it is only natural that we can also obtain the adjoint of an operator
A:H\toE
H
E
A*:E*\toH
A*f=hf
\langlehf,h\rangleH=f(Ah).
Let
\left(E,\| ⋅ \|E\right),\left(F,\| ⋅ \|F\right)
A:D(A)\toF
D(A)\subsetE
A
D(A)
E
A*
D\left(A*\right):=\left\{g\inF*:~\existsc\geq0:~forallu\inD(A):~|g(Au)|\leqc ⋅ \|u\|E\right\}.
Now for arbitrary but fixed
g\inD(A*)
f:D(A)\to\R
f(u)=g(Au)
g
D(A*)
D(A)
|f(u)|=|g(Au)|\leqc ⋅ \|u\|E
f
\hat{f}
E
A*
D\left(A*\right)\toE*
D\left(A*\right)\to(D(A))*.
A
E
g\inD\left(A*\right)
Now, we can define the adjoint of
A
\begin{align} A*:F*\supsetD(A*)&\toE*\\ g&\mapstoA*g=\hatf. \end{align}
The fundamental defining identity is thus
g(Au)=\left(A*g\right)(u)
u\inD(A).
\langle ⋅ , ⋅ \rangle
\langleAx,y\rangle=\left\langlex,A*y\right\rangle forallx,y\inH.
Existence and uniqueness of this operator follows from the Riesz representation theorem.
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.
The following properties of the Hermitian adjoint of bounded operators are immediate:
If we define the operator norm of by
\|A\|op:=\sup\left\{\|Ax\|:\|x\|\le1\right\}
then
\left\|A*\right\|op=\|A\|op.
Moreover,
\left\|A*A\right\|op=
2. | |
\|A\| | |
op |
One says that a norm that satisfies this condition behaves like a "largest value", extrapolating from the case of self-adjoint operators.
The set of bounded linear operators on a complex Hilbert space together with the adjoint operation and the operator norm form the prototype of a C*-algebra.
Let the inner product
\langle ⋅ , ⋅ \rangle
\langleAx,y\rangle=\langlex,z\rangle forallx\inD(A).
Owing to the density of
D(A)
z
A*y=z.
Properties 1.–5. hold with appropriate clauses about domains and codomains. For instance, the last property now states that is an extension of if, and are densely defined operators.
For every
y\in\kerA*,
x\mapsto\langleAx,y\rangle=\langlex,A*y\rangle
y\in(\operatorname{im}A)\perp.
Conversely, the assumption that
y\in(\operatorname{im}A)\perp
x\mapsto\langleAx,y\rangle
A*
y\inD(A*).
x\inD(A),
\langleAx,y\rangle=\langlex,A*y\rangle=0
A*y\inD(A)\perp=\overline{D(A)}\perp=\{0\},
D(A)
This property shows that
\operatorname{ker}A*
D(A*)
If
H1
H2
H1 ⊕ H2
l\langle(a,b),(c,d)r
\rangle | |
H1 ⊕ H2 |
\stackrel{def
where
a,c\inH1
b,d\inH2.
Let
J\colonH ⊕ H\toH ⊕ H
J(\xi,η)=(-η,\xi).
G(A*)=\{(x,y)\midx\inD(A*), y=A*x\}\subseteqH ⊕ H
A*
JG(A):
G(A*)=(JG(A))\perp=\{(x,y)\inH ⊕ H:l\langle(x,y),(-A\xi,\xi)r\rangleH=0 \forall\xi\inD(A)\}.
The assertion follows from the equivalences
l\langle(x,y),(-A\xi,\xi)r\rangle=0 \Leftrightarrow \langleA\xi,x\rangle=\langle\xi,y\rangle,
and
l[\forall\xi\inD(A) \langleA\xi,x\rangle=\langle\xi,y\rangler] \Leftrightarrow x\inD(A*) \& y=A*x.
An operator
A
G(A)
H ⊕ H.
G(A*)
A*
An operator
A
Gcl(A)\subseteqH ⊕ H
G(A)
Gcl(A)
A
(0,v)\notinGcl(A)
v=0.
The adjoint
A*
A
v\inH,
v\inD(A*)\perp \Leftrightarrow (0,v)\inGcl(A),
which, in turn, is proven through the following chain of equivalencies:
\begin{align} v\inD(A*)\perp&\Longleftrightarrow(v,0)\inG(A*)\perp\Longleftrightarrow(v,0)\in(JG(A))cl=JGcl(A)\\ &\Longleftrightarrow(0,-v)=J-1(v,0)\inGcl(A)\\ &\Longleftrightarrow(0,v)\inGcl(A). \end{align}
The closure
Acl
A
Gcl(A)
A**=Acl,
G(A**)=Gcl(A).
To prove this, observe that
J*=-J,
\langleJx,y\rangleH=-\langlex,Jy\rangleH,
x,y\inH ⊕ H.
\begin{align} \langleJ(x1,x2),(y1,y2)\rangleH&=\langle(-x2,x1),(y1,y2)\rangleH=\langle-x2,y1\rangleH+\langlex1,y2\rangleH\\ &=\langlex1,y2\rangleH+\langlex2,-y1\rangleH =\langle(x1,x2),-J(y1,y2)\rangleH. \end{align}
y\inH ⊕ H
V\subseteqH ⊕ H,
y\in(JV)\perp
Jy\inV\perp.
J[(JV)\perp]=V\perp
[J[(JV)\perp]]\perp=Vcl.
V=G(A),
Gcl(A)=G(A**).
For a closable operator
A,
A*=\left(Acl\right)*,
G(A*)=G\left(\left(Acl\right)*\right).
G\left(\left(Acl\right)*\right)=\left(JGcl(A)\right)\perp=\left(\left(JG(A)\right)cl\right)\perp=(JG(A))\perp=G(A*).
Let
H=L2(R,l),
l
f\notinL2,
\varphi0\inL2\setminus\{0\}.
A\varphi=\langlef,\varphi\rangle\varphi0.
It follows that
D(A)=\{\varphi\inL2\mid\langlef,\varphi\rangle ≠ infty\}.
D(A)
L2
1[-n,n] ⋅ \varphi \stackrel{L2}{\to} \varphi,
A
\varphi\inD(A)
\psi\inD(A*),
\langle\varphi,A*\psi\rangle=\langleA\varphi,\psi\rangle=\langle\langlef,\varphi\rangle\varphi0,\psi\rangle=\langlef,\varphi\rangle ⋅ \langle\varphi0,\psi\rangle=\langle\varphi,\langle\varphi0,\psi\ranglef\rangle.
Thus,
A*\psi=\langle\varphi0,\psi\ranglef.
ImA*\subseteqH=L2.
f\notinL2,
\langle\varphi0,\psi\rangle=0.
D(A*)=
\perp. | |
\{\varphi | |
0\} |
A*
D(A*).
A
A**.
A bounded operator is called Hermitian or self-adjoint if
A=A*
which is equivalent to
\langleAx,y\rangle=\langlex,Ay\rangleforallx,y\inH.
In some sense, these operators play the role of the real numbers (being equal to their own "complex conjugate") and form a real vector space. They serve as the model of real-valued observables in quantum mechanics. See the article on self-adjoint operators for a full treatment.
For a conjugate-linear operator the definition of adjoint needs to be adjusted in order to compensate for the complex conjugation. An adjoint operator of the conjugate-linear operator on a complex Hilbert space is an conjugate-linear operator with the property:
\langleAx,y\rangle=\overline{\left\langlex,A*y\right\rangle} forallx,y\inH.
The equation
\langleAx,y\rangle=\left\langlex,A*y\right\rangle
is formally similar to the defining properties of pairs of adjoint functors in category theory, and this is where adjoint functors got their name from.