Hermitian adjoint

Consider a linear map ${\displaystyle A:H_{1}\to H_{2}}$  between Hilbert spaces. Without taking care of any details, the adjoint operator is the (in most cases uniquely defined) linear operator ${\displaystyle A^{*}:H_{2}\to H_{1}}$  fulfilling

${\displaystyle \left\langle Ah_{1},h_{2}\right\rangle _{H_{2}}=\left\langle h_{1},A^{*}h_{2}\right\rangle _{H_{1}},}$ 

where ${\displaystyle \langle \cdot ,\cdot \rangle _{H_{i}}}$  is the inner product in the Hilbert space ${\displaystyle H_{i}}$ , which is linear in the first coordinate and conjugate linear in the second coordinate. Note the special case where both Hilbert spaces are identical and ${\displaystyle A}$  is an operator on that Hilbert space.

When one trades the inner product for the dual pairing, one can define the adjoint, also called the transpose, of an operator ${\displaystyle A:E\to F}$ , where ${\displaystyle E,F}$  are Banach spaces with corresponding norms ${\displaystyle \|\cdot \|_{E},\|\cdot \|_{F}}$ . Here (again not considering any technicalities), its adjoint operator is defined as ${\displaystyle A^{*}:F^{*}\to E^{*}}$  with

${\displaystyle A^{*}f=f\circ A:u\mapsto f(Au),}$ 

i.e., ${\displaystyle \left(A^{*}f\right)(u)=f(Au)}$  for ${\displaystyle f\in F^{*},u\in E}$ .

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 Riesz representation theorem). Then it is only natural that we can also obtain the adjoint of an operator ${\displaystyle A:H\to E}$ , where ${\displaystyle H}$  is a Hilbert space and ${\displaystyle E}$  is a Banach space. The dual is then defined as ${\displaystyle A^{*}:E^{*}\to H}$  with ${\displaystyle A^{*}f=h_{f}}$  such that

${\displaystyle \langle h_{f},h\rangle _{H}=f(Ah).}$ 

Let ${\displaystyle \left(E,\|\cdot \|_{E}\right),\left(F,\|\cdot \|_{F}\right)}$  be Banach spaces. Suppose ${\displaystyle A:D(A)\to F}$  and ${\displaystyle D(A)\subset E}$ , and suppose that ${\displaystyle A}$  is a (possibly unbounded) linear operator which is densely defined (i.e., ${\displaystyle D(A)}$  is dense in ${\displaystyle E}$ ). Then its adjoint operator ${\displaystyle A^{*}}$  is defined as follows. The domain is

${\displaystyle D\left(A^{*}\right):=\left\{g\in F^{*}:~\exists c\geq 0:~{\mbox{ for all }}u\in D(A):~|g(Au)|\leq c\cdot \|u\|_{E}\right\}.}$ 

Now for arbitrary but fixed ${\displaystyle g\in D(A^{*})}$  we set ${\displaystyle f:D(A)\to \mathbb {R} }$  with ${\displaystyle f(u)=g(Au)}$ . By choice of ${\displaystyle g}$  and definition of ${\displaystyle D(A^{*})}$ , f is (uniformly) continuous on ${\displaystyle D(A)}$  as ${\displaystyle |f(u)|=|g(Au)|\leq c\cdot \|u\|_{E}}$ . Then by the Hahn–Banach theorem, or alternatively through extension by continuity, this yields an extension of ${\displaystyle f}$ , called ${\displaystyle {\hat {f}}}$ , defined on all of ${\displaystyle E}$ . This technicality is necessary to later obtain ${\displaystyle A^{*}}$  as an operator ${\displaystyle D\left(A^{*}\right)\to E^{*}}$  instead of ${\displaystyle D\left(A^{*}\right)\to (D(A))^{*}.}$  Remark also that this does not mean that ${\displaystyle A}$  can be extended on all of ${\displaystyle E}$  but the extension only worked for specific elements ${\displaystyle g\in D\left(A^{*}\right)}$ .

Now, we can define the adjoint of ${\displaystyle A}$  as

${\displaystyle {\begin{aligned}A^{*}:F^{*}\supset D(A^{*})&\to E^{*}\\g&\mapsto A^{*}g={\hat {f}}.\end{aligned}}}$ 

The fundamental defining identity is thus

${\displaystyle g(Au)=\left(A^{*}g\right)(u)}$  for ${\displaystyle u\in D(A).}$ 

Suppose H is a complex Hilbert space, with inner product ${\displaystyle \langle \cdot ,\cdot \rangle }$ . Consider a continuous linear operator A : H → H (for linear operators, continuity is equivalent to being a bounded operator). Then the adjoint of A is the continuous linear operator A^∗ : H → H satisfying

${\displaystyle \langle Ax,y\rangle =\left\langle x,A^{*}y\right\rangle \quad {\mbox{for all }}x,y\in H.}$ 

Existence and uniqueness of this operator follows from the Riesz representation theorem.^[2]

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:^[2]

1. Involutivity: A^∗∗ = A
2. If A is invertible, then so is A^∗, with ${\textstyle \left(A^{*}\right)^{-1}=\left(A^{-1}\right)^{*}}$ 
3. Conjugate linearity:
   • (A + B)^∗ = A^∗ + B^∗
   • (λA)^∗ = λA^∗, where λ denotes the complex conjugate of the complex number λ
4. "Anti-distributivity": (AB)^∗ = B^∗A^∗

If we define the operator norm of A by

${\displaystyle \|A\|_{\text{op}}:=\sup \left\{\|Ax\|:\|x\|\leq 1\right\}}$ 

then

${\displaystyle \left\|A^{*}\right\|_{\text{op}}=\|A\|_{\text{op}}.}$ ^[2]

Moreover,

${\displaystyle \left\|A^{*}A\right\|_{\text{op}}=\|A\|_{\text{op}}^{2}.}$ ^[2]

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 H together with the adjoint operation and the operator norm form the prototype of a C*-algebra.

Let the inner product ${\displaystyle \langle \cdot ,\cdot \rangle }$  be linear in the first argument. A densely defined operator A from a complex Hilbert space H to itself is a linear operator whose domain D(A) is a dense linear subspace of H and whose values lie in H.^[3] By definition, the domain D(A^∗) of its adjoint A^∗ is the set of all y ∈ H for which there is a z ∈ H satisfying

${\displaystyle \langle Ax,y\rangle =\langle x,z\rangle \quad {\mbox{for all }}x\in D(A).}$ 

Owing to the density of ${\displaystyle D(A)}$  and Riesz representation theorem, ${\displaystyle z}$  is uniquely defined, and, by definition, ${\displaystyle A^{*}y=z.}$ ^[4]

Properties 1.–5. hold with appropriate clauses about domains and codomains.^{[clarification needed]} For instance, the last property now states that (AB)^∗ is an extension of B^∗A^∗ if A, B and AB are densely defined operators.^[5]

For every ${\displaystyle y\in \ker A^{*},}$  the linear functional ${\displaystyle x\mapsto \langle Ax,y\rangle =\langle x,A^{*}y\rangle }$  is identically zero, and hence ${\displaystyle y\in (\operatorname {im} A)^{\perp }.}$ 

Conversely, the assumption that ${\displaystyle y\in (\operatorname {im} A)^{\perp }}$  causes the functional ${\displaystyle x\mapsto \langle Ax,y\rangle }$  to be identically zero. Since the functional is obviously bounded, the definition of ${\displaystyle A^{*}}$  assures that ${\displaystyle y\in D(A^{*}).}$  The fact that, for every ${\displaystyle x\in D(A),}$  ${\displaystyle \langle Ax,y\rangle =\langle x,A^{*}y\rangle =0}$  shows that ${\displaystyle A^{*}y\in D(A)^{\perp }={\overline {D(A)}}^{\perp }=\{0\},}$  given that ${\displaystyle D(A)}$  is dense.

This property shows that ${\displaystyle \operatorname {ker} A^{*}}$  is a topologically closed subspace even when ${\displaystyle D(A^{*})}$  is not.

If ${\displaystyle H_{1}}$  and ${\displaystyle H_{2}}$  are Hilbert spaces, then ${\displaystyle H_{1}\oplus H_{2}}$  is a Hilbert space with the inner product

${\displaystyle {\bigl \langle }(a,b),(c,d){\bigr \rangle }_{H_{1}\oplus H_{2}}{\stackrel {\text{def}}{=}}\langle a,c\rangle _{H_{1}}+\langle b,d\rangle _{H_{2}},}$ 

where ${\displaystyle a,c\in H_{1}}$  and ${\displaystyle b,d\in H_{2}.}$ 

Let ${\displaystyle J\colon H\oplus H\to H\oplus H}$  be the symplectic mapping, i.e. ${\displaystyle J(\xi ,\eta )=(-\eta ,\xi ).}$  Then the graph

${\displaystyle G(A^{*})=\{(x,y)\mid x\in D(A^{*}),\ y=A^{*}x\}\subseteq H\oplus H}$ 

of ${\displaystyle A^{*}}$  is the orthogonal complement of ${\displaystyle JG(A):}$ 

${\displaystyle G(A^{*})=(JG(A))^{\perp }=\{(x,y)\in H\oplus H:{\bigl \langle }(x,y),(-A\xi ,\xi ){\bigr \rangle }_{H\oplus H}=0\;\;\forall \xi \in D(A)\}.}$ 

The assertion follows from the equivalences

${\displaystyle {\bigl \langle }(x,y),(-A\xi ,\xi ){\bigr \rangle }=0\quad \Leftrightarrow \quad \langle A\xi ,x\rangle =\langle \xi ,y\rangle ,}$ 

and

${\displaystyle {\Bigl [}\forall \xi \in D(A)\ \ \langle A\xi ,x\rangle =\langle \xi ,y\rangle {\Bigr ]}\quad \Leftrightarrow \quad x\in D(A^{*})\ \&\ y=A^{*}x.}$ 

An operator ${\displaystyle A}$  is closed if the graph ${\displaystyle G(A)}$  is topologically closed in ${\displaystyle H\oplus H.}$  The graph ${\displaystyle G(A^{*})}$  of the adjoint operator ${\displaystyle A^{*}}$  is the orthogonal complement of a subspace, and therefore is closed.

An operator ${\displaystyle A}$  is closable if the topological closure ${\displaystyle G^{\text{cl}}(A)\subseteq H\oplus H}$  of the graph ${\displaystyle G(A)}$  is the graph of a function. Since ${\displaystyle G^{\text{cl}}(A)}$  is a (closed) linear subspace, the word "function" may be replaced with "linear operator". For the same reason, ${\displaystyle A}$  is closable if and only if ${\displaystyle (0,v)\notin G^{\text{cl}}(A)}$  unless ${\displaystyle v=0.}$ 

The adjoint ${\displaystyle A^{*}}$  is densely defined if and only if ${\displaystyle A}$  is closable. This follows from the fact that, for every ${\displaystyle v\in H,}$ 

${\displaystyle v\in D(A^{*})^{\perp }\ \Leftrightarrow \ (0,v)\in G^{\text{cl}}(A),}$ 

which, in turn, is proven through the following chain of equivalencies:

${\displaystyle {\begin{aligned}v\in D(A^{*})^{\perp }&\Longleftrightarrow (v,0)\in G(A^{*})^{\perp }\Longleftrightarrow (v,0)\in (JG(A))^{\text{cl}}=JG^{\text{cl}}(A)\\&\Longleftrightarrow (0,-v)=J^{-1}(v,0)\in G^{\text{cl}}(A)\\&\Longleftrightarrow (0,v)\in G^{\text{cl}}(A).\end{aligned}}}$ 

The closure ${\displaystyle A^{\text{cl}}}$  of an operator ${\displaystyle A}$  is the operator whose graph is ${\displaystyle G^{\text{cl}}(A)}$  if this graph represents a function. As above, the word "function" may be replaced with "operator". Furthermore, ${\displaystyle A^{**}=A^{\text{cl}},}$  meaning that ${\displaystyle G(A^{**})=G^{\text{cl}}(A).}$ 

To prove this, observe that ${\displaystyle J^{*}=-J,}$  i.e. ${\displaystyle \langle Jx,y\rangle _{H\oplus H}=-\langle x,Jy\rangle _{H\oplus H},}$  for every ${\displaystyle x,y\in H\oplus H.}$  Indeed,

${\displaystyle {\begin{aligned}\langle J(x_{1},x_{2}),(y_{1},y_{2})\rangle _{H\oplus H}&=\langle (-x_{2},x_{1}),(y_{1},y_{2})\rangle _{H\oplus H}=\langle -x_{2},y_{1}\rangle _{H}+\langle x_{1},y_{2}\rangle _{H}\\&=\langle x_{1},y_{2}\rangle _{H}+\langle x_{2},-y_{1}\rangle _{H}=\langle (x_{1},x_{2}),-J(y_{1},y_{2})\rangle _{H\oplus H}.\end{aligned}}}$ 

In particular, for every ${\displaystyle y\in H\oplus H}$  and every subspace ${\displaystyle V\subseteq H\oplus H,}$  ${\displaystyle y\in (JV)^{\perp }}$  if and only if ${\displaystyle Jy\in V^{\perp }.}$  Thus, ${\displaystyle J[(JV)^{\perp }]=V^{\perp }}$  and ${\displaystyle [J[(JV)^{\perp }]]^{\perp }=V^{\text{cl}}.}$  Substituting ${\displaystyle V=G(A),}$  obtain ${\displaystyle G^{\text{cl}}(A)=G(A^{**}).}$ 

For a closable operator ${\displaystyle A,}$  ${\displaystyle A^{*}=\left(A^{\text{cl}}\right)^{*},}$  meaning that ${\displaystyle G(A^{*})=G\left(\left(A^{\text{cl}}\right)^{*}\right).}$  Indeed,

${\displaystyle G\left(\left(A^{\text{cl}}\right)^{*}\right)=\left(JG^{\text{cl}}(A)\right)^{\perp }=\left(\left(JG(A)\right)^{\text{cl}}\right)^{\perp }=(JG(A))^{\perp }=G(A^{*}).}$ 

Let ${\displaystyle H=L^{2}(\mathbb {R} ,l),}$  where ${\displaystyle l}$  is the linear measure. Select a measurable, bounded, non-identically zero function ${\displaystyle f\notin L^{2},}$  and pick ${\displaystyle \varphi _{0}\in L^{2}\setminus \{0\}.}$  Define

${\displaystyle A\varphi =\langle f,\varphi \rangle \varphi _{0}.}$ 

It follows that ${\displaystyle D(A)=\{\varphi \in L^{2}\mid \langle f,\varphi \rangle \neq \infty \}.}$  The subspace ${\displaystyle D(A)}$  contains all the ${\displaystyle L^{2}}$  functions with compact support. Since ${\displaystyle \mathbf {1} _{[-n,n]}\cdot \varphi \ {\stackrel {L^{2}}{\to }}\ \varphi ,}$  ${\displaystyle A}$  is densely defined. For every ${\displaystyle \varphi \in D(A)}$  and ${\displaystyle \psi \in D(A^{*}),}$ 

${\displaystyle \langle \varphi ,A^{*}\psi \rangle =\langle A\varphi ,\psi \rangle =\langle \langle f,\varphi \rangle \varphi _{0},\psi \rangle =\langle f,\varphi \rangle \cdot \langle \varphi _{0},\psi \rangle =\langle \varphi ,\langle \varphi _{0},\psi \rangle f\rangle .}$ 

Thus, ${\displaystyle A^{*}\psi =\langle \varphi _{0},\psi \rangle f.}$  The definition of adjoint operator requires that ${\displaystyle \mathop {\text{Im}} A^{*}\subseteq H=L^{2}.}$  Since ${\displaystyle f\notin L^{2},}$  this is only possible if ${\displaystyle \langle \varphi _{0},\psi \rangle =0.}$  For this reason, ${\displaystyle D(A^{*})=\{\varphi _{0}\}^{\perp }.}$  Hence, ${\displaystyle A^{*}}$  is not densely defined and is identically zero on ${\displaystyle D(A^{*}).}$  As a result, ${\displaystyle A}$  is not closable and has no second adjoint ${\displaystyle A^{**}.}$ 

A bounded operator A : H → H is called Hermitian or self-adjoint if

${\displaystyle A=A^{*}}$ 

which is equivalent to

${\displaystyle \langle Ax,y\rangle =\langle x,Ay\rangle {\mbox{ for all }}x,y\in H.}$ ^[6]

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 A on a complex Hilbert space H is an conjugate-linear operator A^∗ : H → H with the property:

${\displaystyle \langle Ax,y\rangle ={\overline {\left\langle x,A^{*}y\right\rangle }}\quad {\text{for all }}x,y\in H.}$ 

The equation

${\displaystyle \langle Ax,y\rangle =\left\langle x,A^{*}y\right\rangle }$ 

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.

• Mathematical concepts
  • Conjugate transpose
  • Hermitian operator
  • Pullback § Functional analysis
  • Transpose of linear maps
• Physical applications
  • Operator (physics)
  • †-algebra

• Brezis, Haim (2011), Functional Analysis, Sobolev Spaces and Partial Differential Equations (first ed.), Springer, ISBN 978-0-387-70913-0.
• Reed, Michael; Simon, Barry (2003), Functional Analysis, Elsevier, ISBN 981-4141-65-8.
• 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.