In mathematics, specifically in functional analysis, each bounded linear operator on a complex Hilbert space has a corresponding 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 there can be defined an adjoint operator for linear (and possibly unbounded) operators between Banach spaces.

The adjoint of an operator A may also be called the Hermitian adjoint, Hermitian conjugate or Hermitian transpose[1] (after Charles Hermite) of A and is denoted by A or A (the latter especially when used in conjunction with the bra–ket notation).

## Informal definition

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

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

where ${\displaystyle \langle \cdot ,\cdot \rangle _{H_{i}}}$ is the inner product in the Hilbert space ${\displaystyle H_{i}}$. Note the special case where both Hilbert spaces are identical and ${\displaystyle A}$ 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 ${\displaystyle A:E\to F}$, 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. ${\displaystyle (A^{*}f)(u)=f(Au)}$ 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

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

## Definition for unbounded operators between normed spaces

Let ${\displaystyle (E,\|\cdot \|_{E}),(F,\|\cdot \|_{F})}$ be Banach spaces. Suppose ${\displaystyle A:E\supset D(A)\to F}$ 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

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

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 ${\displaystyle D(A^{*})\to E^{*}}$ instead of ${\displaystyle D(A^{*})\to (D(A))^{*}.}$ 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 ${\displaystyle g\in D(A^{*})}$.

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

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

## Definition for bounded operators between Hilbert spaces

Suppose H is a complex Hilbert space, with inner product ${\displaystyle \langle \cdot ,\cdot \rangle }$. 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

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

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.

## Properties

The following properties of the Hermitian adjoint of bounded operators are immediate:[2]

1. A∗∗ = Ainvolutiveness
2. If A is invertible, then so is A, with (A)−1 = (A−1)
3. (A + B) = A + B
4. (λA) = λA, where λ denotes the complex conjugate of the complex number λantilinearity (together with 3.)
5. (AB) = BA

If we define the operator norm of A by

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

then

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

Moreover,

${\displaystyle \|A^{*}A\|_{op}=\|A\|_{op}^{2}.}$[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.

## 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 lies in H.[3] By definition, the domain D(A) of its adjoint A is the set of all yH for which there is a zH 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.[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.[5]

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

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

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][clarification needed]

{\displaystyle {\begin{aligned}A^{*}x=0&\iff \langle A^{*}x,y\rangle =0\quad \forall y\in H\\&\iff \langle x,Ay\rangle =0\quad \forall 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.[clarification needed]

## Hermitian operators

A bounded operator A : HH 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 : HH with the property:

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

The equation

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

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. ^ David A. B. Miller (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

## References

• Reed, Michael; Simon, Barry (2003), Functional Analysis, Elsevier, ISBN 981-4141-65-8.
• Rudin, Walter (1991), Functional Analysis (second ed.), McGraw-Hill, ISBN 0-07-054236-8.
• Brezis, Haim (2011), Functional Analysis, Sobolev Spaces and Partial Differential Equations (first ed.), Springer, ISBN 978-0-387-70913-0.