Hermitian adjoint

Consider a linear operator ${\displaystyle A:H_{1}\to H_{2}}$ 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 antilinear 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 dual pairing for the inner product, 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

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

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

Note that 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 ${\displaystyle A:H\to E}$, 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 Hahn–Banach theorem or alternatively through extension by continuity this yields an extension of ${\displaystyle f}$, called ${\displaystyle {\hat {f}}}$ defined on all of ${\displaystyle E}$. Note that 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. Anti-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.

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

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

and A^∗(y) is defined to be the z thus found.[4]

Properties 1.–5. hold with appropriate clauses about domains and codomains. 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]

The relationship between the image of A and the kernel of its adjoint is given by:

${\displaystyle {\begin{aligned}\ker A^{*}&=\left(\operatorname {im} \ A\right)^{\bot }\\\left(\ker A^{*}\right)^{\bot }&={\overline {\operatorname {im} \ A}}\end{aligned}}}$

These statements are equivalent. See orthogonal complement for the proof of this and for the definition of ${\displaystyle \bot }$.

Proof of the first equation:[6]

${\displaystyle {\begin{aligned}A^{*}x=0&\iff \left\langle A^{*}x,y\right\rangle =0\quad {\mbox{ for all }}y\in H\\&\iff \left\langle x,Ay\right\rangle =0\quad {\mbox{ for all }}y\in H\\&\iff x\ \bot \ \operatorname {im} \ A\end{aligned}}}$

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 operator[7] always is.

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

${\displaystyle A=A^{*}}$

which is equivalent to

${\displaystyle \langle Ax,y\rangle =\langle x,Ay\rangle {\mbox{ for all }}x,y\in H.}$[8]

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 an 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 A on a complex Hilbert space H is an antilinear 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
  • Hermitian operator
  • Norm (mathematics)
  • 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. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.