In mathematics, specifically in functional analysis, one associates to every linear operator on a Hilbert space its 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 is also sometimes called the Hermitian adjoint of A and is denoted by A^* or (the latter especially when used in conjunction with the bra-ket notation).

Contents

1 Definition for bounded operators 2 Properties 3 Hermitian operators 4 Adjoints of unbounded operators 5 Other adjoints 6 See also

Definition for bounded operators

Suppose H is a Hilbert space, with inner product <.,.>. 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:

This operator A* is the adjoint of A.

Properties

Immediate properties:

1. A** = A
2. (A + B )* = A* + B*
3. (λA)* = λ* A*, where λ* denotes the complex conjugate of the complex number λ
4. (AB)* = B* A*

If we define the operator norm of A by

then

.

Moreover,

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-star algebra.

Hermitian operators

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

A = A*

which is equivalent to

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

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
• Physical application
  • Dual space
  • Bra-ket notation
  • Quantum mechanics
  • observable