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 is also sometimes called the Hermitian conjugate (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 (note that for linear operators, continuity is equivalent to being a bounded operator).

Using the Riesz representation theorem, one can show that there exists a unique continuous linear operator A* : HH with the following property:

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

This operator A* is the adjoint of A. 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

Following properties of Hermitian adjoint on bounded operators are immediate:

1. A** = A – involutiveness
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 λ – antilinearity (together with 3.)
5. (AB)* = B* A*

If we define the operator norm of A by

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

then

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

Moreover,

$\| A^* A \| _{op} = \| A \| _{op}^2$ (C*-norm)

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.

Hermitian adjoint is defined for any densely defined operator,[1] not necessarily a bounded one. Properties 1.–5. hold with appropriate clauses about domains and codomains. 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:

\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[2] always is.

## Hermitian operators

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

$A = A^{*}$

which is equivalent to

$\lang Ax , y \rang = \lang x , A y \rang \mbox{ for all } x,y\in H.$

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 Hilbert space H is an antilinear operator A* : HH with the property:

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

The equation

$\lang Ax , y \rang = \lang x , A^* y \rang$

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.