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 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).

## Definition for bounded operators

Suppose H is a 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

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

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. A)* = λA*, where λ denotes the complex conjugate of the complex number λantilinearity (together with 3.)
5. (AB)* = B* A*

If we define the operator norm of A by

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

then

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

Moreover,

$\| A^* A \| _{op} = \| A \| _{op}^2.$[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 H 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 A on a Hilbert space H is a linear operator whose domain D(A) is a dense linear subspace of H and whose co-domain is H.[3] Its adjoint A* has as domain D(A*) 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) equals the z 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 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:

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

Proof of the first equation:[6]

\begin{align} 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{align}

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

$A = A^{*}$

which is equivalent to

$\langle Ax , y \rangle = \langle x , A y \rangle \mbox{ for all } x,y\in H.$[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 Hilbert space H is an antilinear operator A* : HH with the property:

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

The equation

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

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, pp. 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.