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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).
Definition for bounded operators
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.
- A** = A
- (A + B )* = A* + B*
- (λA)* = 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
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.
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.
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