Hermitian adjoint

In mathematics, specifically in functional analysis, each 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 adjoint (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 $langlecdot,cdot\; angle$. Consider a continuous linear operator "A" : "H" → "H" (this is the same as a bounded operator).

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

: $lang\; Ax\; ,\; y\; ang\; =\; lang\; x\; ,\; A^*\; y\; ang\; quad\; mbox\{for\; all\; \}\; x,yin\; H.$

This operator "A"* is the adjoint of "A".

This can be seen as a generalization of the conjugate transpose or "adjoint" matrix of a square matrix which has a similar property involving the standard complex inner product.

Properties

Immediate properties:
# "A"** = "A"
# If "A" is invertible, so is "A"*. Then, ("A"*)⁻¹ = ("A"⁻¹)*
# ("A" + "B")* = "A"* + "B"*
# (λ"A")* = λ* "A"*, where λ* denotes the complex conjugate of the complex number λ
# ("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,:$|\; A^*\; A\; |\; \_\{op\}\; =\; |\; A\; |\; \_\{op\}^2$

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.

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

Proof of the first equation::

The second equation follows from the first by taking the orthogonal space on both sides. Note that in general, the image need not be closed, but the kernel of a continuous operator always is.

Hermitian operators

A bounded operator "A" : "H" → "H" is called Hermitian or self-adjoint if : "A" = "A"*which is equivalent to:$lang\; Ax\; ,\; y\; ang\; =\; lang\; x\; ,\; A\; y\; ang\; mbox\{\; for\; all\; \}\; x,yin\; H.$

In some sense, these operators play the role of the real numbers (being equal to their own "complex conjugate"). They serve as the model of real-valued observables in quantum mechanics. See the article on self-adjoint operators for a full treatment.

Adjoints of unbounded operators

Many operators of importance are not continuous and are only defined on a subspace of a Hilbert space. In this situation, one may still define an adjoint, as is explained in the article on self-adjoint operators.

Other adjoints

The equation: $lang\; Ax\; ,\; y\; ang\; =\; lang\; x\; ,\; A^*\; y\; ang$is formally similar to the defining properties of pairs of adjoint functors in category theory, and this is where adjoint functors got their name.

See also

* Mathematical concepts
** Linear algebra
** Inner product
** Hilbert space
** Hermitian operator
** Norm
** Operator norm
** Transpose of a linear maps
* Physical applications
** Dual space
** Bra-ket notation
** Quantum mechanics
** Observables