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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' of ''A'' and is denoted by ''A''
^★ or ''A''^† (the latter especially when used in conjunction with the bra-ket notation).

Contents

Definition for bounded operators

Properties

Hermitian operators

Adjoints of unbounded operators

Other adjoints

See also

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''.

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:
:$ker\; A^$
★ = left( operatorname{im} A
ight)^ot
:$left(\; ker\; A^$
★
ight)^ot = overline{operatorname{im} A}
Proof of the first equation:
:
A^
★ x = 0 &iff
langle A^
★ x,y
angle = 0 quad orall y in H \ &iff
langle x,Ay
angle = 0 quad orall y in H \ &iff
x ot operatorname{im} A
end{align}
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

★ Physical application

★
★ Dual space

★
★ Bra-ket notation

★
★ Quantum mechanics

★
★ observable