In
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, specifically in
functional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limitrelated structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, each linear operator on a
Hilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the twodimensional Euclidean plane and threedimensional space to spaces with any finite or infinite number of dimensions...
has a corresponding
adjoint operator.
Adjoints of operators generalize
conjugate transposeIn mathematics, the conjugate transpose, Hermitian transpose, Hermitian conjugate, or adjoint matrix of an mbyn matrix A with complex entries is the nbym matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry...
s of square matrices to (possibly) infinitedimensional 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 conjugateIn mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...
of a complex number.
The adjoint of an operator
A is also sometimes called the
Hermitian conjugate (after
Charles HermiteCharles Hermite was a French mathematician who did research on number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra....
) of
A and is denoted by
A^{*} or
A^{†} (the latter especially when used in conjunction with the
braket notationBraket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics...
).
Definition for bounded operators
Suppose
H is a
Hilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the twodimensional Euclidean plane and threedimensional space to spaces with any finite or infinite number of dimensions...
, with inner product
. Consider a continuous linear operator
A :
H →
H (this is the same as a
bounded operatorIn functional analysis, a branch of mathematics, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L to that of v is bounded by the same number, over all nonzero vectors v in X...
).
Using the
Riesz representation theoremThere are several wellknown theorems in functional analysis known as the Riesz representation theorem. They are named in honour of Frigyes Riesz. The Hilbert space 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.This can be seen as a generalization of the
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, then so is A*, with (A*)^{−1} = (A^{−1})*
 (A + B)* = A* + B*
 (λA)* = λ* A*, where λ* denotes the complex conjugate
In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...
of the complex numberA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the onedimensional number line to the twodimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
λ
 (AB)* = B*A*
If we define the
operator normIn mathematics, the operator norm is a means to measure the "size" of certain linear operators. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Introduction and definition :...
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* algebra.
The relationship between the image of
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
selfadjointIn mathematics, an element x of a staralgebra is selfadjoint if x^*=x.A collection C of elements of a staralgebra is selfadjoint if it is closed under the involution operation...
if
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 realvalued
observableIn physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value off...
s in
quantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particlelike and wavelike behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
. See the article on
selfadjoint operatorIn mathematics, on a finitedimensional inner product space, a selfadjoint operator is an operator that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose...
s for a full treatment.
Adjoints of unbounded operators
Many operators of importance are not continuous and are only defined on a
subspaceIn mathematics:* Euclidean subspace, in linear algebra, a set of vectors in ndimensional Euclidean space that is closed under addition and scalar multiplication...
of a Hilbert space. In this situation, one may still define an adjoint, as is explained in the articles on
selfadjoint operatorIn mathematics, on a finitedimensional inner product space, a selfadjoint operator is an operator that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose...
s and
unbounded operatorIn mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases....
s.
Adjoints of antilinear operators
For an
antilinear operator the definition of adjoint need to be adjusted in order to compensate for the complex conjugation. An adjoint operator of the antilinear operator
A on a Hilbert space
H is an antilinear operator
A* :
H →
H
with the property:

Other adjoints
The equation

is formally similar to the defining properties of pairs of adjoint functors in
category theoryCategory theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, and this is where adjoint functors got their name.
See also
 Mathematical concepts
 Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
 Inner product
 Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the twodimensional Euclidean plane and threedimensional space to spaces with any finite or infinite number of dimensions...
 Hermitian operator
 Norm
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...
 Operator norm
In mathematics, the operator norm is a means to measure the "size" of certain linear operators. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Introduction and definition :...
 Transpose of linear maps
 Physical applications
 Dual space
In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finitedimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
 Braket notation
Braket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics...
 Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particlelike and wavelike behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
 Observable
In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value off...
s