In mathematics, specifically in functional analysis, each bounded linear operator on a complex Hilbert space has a corresponding Hermitian adjoint (or adjoint operator). Adjoints of operators generalize conjugate transposes of square matrices to (possibly) infinite-dimensional situations. If one thinks of operators on a complex Hilbert space as generalized complex numbers, then the adjoint of an operator plays the role of the complex conjugate of a complex number.

In a similar sense, one can define an adjoint operator for linear (and possibly unbounded) operators between Banach spaces.

The adjoint of an operator A may also be called the Hermitian conjugate, Hermitian or Hermitian transpose (after Charles Hermite) of A and is denoted by A or A (the latter especially when used in conjunction with the bra–ket notation). Confusingly, A may also be used to represent the conjugate of A.

## Informal definition

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

$\left\langle Ah_{1},h_{2}\right\rangle _{H_{2}}=\left\langle h_{1},A^{*}h_{2}\right\rangle _{H_{1}},$ where $\langle \cdot ,\cdot \rangle _{H_{i}}$ is the inner product in the Hilbert space $H_{i}$ . Note the special case where both Hilbert spaces are identical and $A$ is an operator on some Hilbert space.

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

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

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 $A:H\to E$ , where $H$ is a Hilbert space and $E$ is a Banach space. The dual is then defined as $A^{*}:E^{*}\to H$ with $A^{*}f=h_{f}$ such that

$\langle h_{f},h\rangle _{H}=f(Ah).$ ## Definition for unbounded operators between normed spaces

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

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

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

Now we can define the adjoint of $A$ as

{\begin{aligned}A^{*}:F^{*}\supset D(A^{*})&\to E^{*}\\g&\mapsto A^{*}g={\hat {f}}\end{aligned}} The fundamental defining identity is thus

$g(Au)=\left(A^{*}g\right)(u)$ for $u\in D(A).$ ## Definition for bounded operators between Hilbert spaces

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

$\langle Ax,y\rangle =\left\langle x,A^{*}y\right\rangle \quad {\mbox{for all }}x,y\in H.$ Existence and uniqueness of this operator follows from the Riesz representation theorem.

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.

## Properties

The following properties of the Hermitian adjoint of bounded operators are immediate:

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:
4. "Anti-distributivity": (AB) = BA

If we define the operator norm of A by

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

$\left\|A^{*}\right\|_{\text{op}}=\|A\|_{\text{op}}.$ Moreover,

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

## Adjoint of densely defined unbounded operators between Hilbert spaces

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. By definition, the domain D(A) of its adjoint A is the set of all yH for which there is a zH satisfying

$\langle Ax,y\rangle =\langle x,z\rangle \quad {\mbox{for all }}x\in D(A),$ and A(y) is defined to be the z thus found.

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 BA if A, B and AB are densely defined operators.

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

{\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 $\bot$ .

Proof of the first equation:[clarification needed]

{\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 always is.[clarification needed]

## Hermitian operators

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

$A=A^{*}$ which is equivalent to

$\langle Ax,y\rangle =\langle x,Ay\rangle {\mbox{ for all }}x,y\in H.$ 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 : HH with the property:

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

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

## Footnotes

1. ^ Miller, David A. B. (2008). Quantum Mechanics for Scientists and Engineers. Cambridge University Press. pp. 262, 280.
2. ^ a b c d Reed & Simon 2003, pp. 186–187; Rudin 1991, §12.9
3. ^ See unbounded operator for details.
4. ^ Reed & Simon 2003, p. 252; Rudin 1991, §13.1
5. ^ Rudin 1991, Thm 13.2
6. ^ See Rudin 1991, Thm 12.10 for the case of bounded operators
7. ^ The same as a bounded operator.
8. ^ Reed & Simon 2003, pp. 187; Rudin 1991, §12.11