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In mathematics, specifically in functional analysis, each bounded linear operator on a 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 Hilbert space as "generalized complex numbers", then the adjoint of an operator plays the role of the complex conjugate of a complex number.

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

## Definition for bounded operators

Suppose Template:Mvar is a 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 Template:Mvar 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 Template:Mvar is invertible, then so is A*, with (A*)−1 = (A−1)*
3. (A + B)* = A* + B*
4. A)* = Template:OverlineA*, where Template:Overline denotes the complex conjugate of the complex number λantilinearity (together with 3.)
5. (AB)* = B* A*

If we define the operator norm of Template:Mvar 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 Hilbert space Template:Mvar together with the adjoint operation and the operator norm form the prototype of a C*-algebra.

## Adjoint of densely defined operators

A densely defined operator Template:Mvar on a Hilbert space Template:Mvar is a linear operator whose domain D(A) is a dense linear subspace of Template:Mvar and whose co-domain is Template:Mvar.[3] Its adjoint A* has as domain D(A*) 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) equals the Template:Mvar defined thus.[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 Template:Mvar, Template:Mvar and Template:Mvar are densely defined operators.[5]

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

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

Proof of the first equation:[6]

{\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.

## 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 Template:Mvar on a Hilbert space Template:Mvar 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

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2. See unbounded operator for details.
3. Template:Harvnb
4. See Template:Harvnb for the case of bounded operators
5. The same as a bounded operator.

## References

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