Hermitian adjoint

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 <math> A^\dagger <math> (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 <.,.>. Consider a continuous linear operator A : HH (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* : HH with the following property:

<math> \lang Ax , y \rang = \lang x , A^* y \rang \quad \mbox{for all } x,y\in H<math>

This operator A* is the adjoint of A.


Immediate properties:

  1. A** = A
  2. (A + B )* = A* + B*
  3. A)* = 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

<math> \| A \| _{op} := \sup \{ \|Ax \| : \| x \| \le 1 \} <math>


<math> \| A^* \| _{op} = \| A \| _{op} <math>.


<math> \| A^* A \| _{op} = \| A \| _{op}^2 <math>

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 : HH is called Hermitian or self-adjoint if

A = A*

which is equivalent to

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

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

<math> \lang Ax , y \rang = \lang x , A^* y \rang <math>

is formally similar to the defining properties of pairs of adjoint functors in category theory, and this is where adjoint functors got their name.

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