In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically in
operator theory, each
linear operator
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
on a
Euclidean vector space defines a Hermitian adjoint (or adjoint) operator
on that space according to the rule
:
where
is the
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
on the vector space.
The adjoint may also be called the Hermitian conjugate or simply the Hermitian after
Charles Hermite
Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra.
Hermi ...
. It is often denoted by in fields like
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, especially when used in conjunction with
bra–ket notation
In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets".
A ket is of the form , v \rangle. Mathema ...
in
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
. In finite dimensions where operators are represented by
matrices, the Hermitian adjoint is given by the
conjugate transpose (also known as the Hermitian transpose).
The above definition of an adjoint operator extends verbatim to
bounded linear operators on
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s
. The definition has been further extended to include unbounded ''
densely defined
In mathematics – specifically, in operator theory – a densely defined operator or partially defined operator is a type of partially defined function. In a topological sense, it is a linear operator that is defined "almost everywhere". ...
'' operators whose domain is topologically
dense
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
in—but not necessarily equal to—
Informal definition
Consider a
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
between
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s. Without taking care of any details, the adjoint operator is the (in most cases uniquely defined) linear operator
fulfilling
:
where
is the
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
in the Hilbert space
, which is linear in the first coordinate and
antilinear in the second coordinate. Note the special case where both Hilbert spaces are identical and
is an operator on that Hilbert space.
When one trades the inner product for the dual pairing, one can define the adjoint, also called the
transpose, of an operator
, where
are
Banach spaces with corresponding
norms . Here (again not considering any technicalities), its adjoint operator is defined as
with
:
I.e.,
for
.
Note that the above definition in the Hilbert space setting is really just an application of the Banach space case when one identifies a Hilbert space with its dual. Then it is only natural that we can also obtain the adjoint of an operator
, where
is a Hilbert space and
is a Banach space. The dual is then defined as
with
such that
:
Definition for unbounded operators between Banach spaces
Let
be
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s. Suppose
and
, and suppose that
is a (possibly unbounded) linear operator which is
densely defined
In mathematics – specifically, in operator theory – a densely defined operator or partially defined operator is a type of partially defined function. In a topological sense, it is a linear operator that is defined "almost everywhere". ...
(i.e.,
is dense in
). Then its adjoint operator
is defined as follows. The domain is
:
.
Now for arbitrary but fixed
we set
with
. By choice of
and definition of
, f is (uniformly) continuous on
as
. Then by
Hahn–Banach theorem or alternatively through extension by continuity this yields an extension of
, called
defined on all of
. Note that this technicality is necessary to later obtain
as an operator
instead of
Remark also that this does not mean that
can be extended on all of
but the extension only worked for specific elements
.
Now we can define the adjoint of
as
:
The fundamental defining identity is thus
:
for
Definition for bounded operators between Hilbert spaces
Suppose is a complex
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
, with
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
. Consider a
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
linear operator
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
(for linear operators, continuity is equivalent to being a
bounded operator). Then the adjoint of is the continuous linear operator satisfying
:
Existence and uniqueness of this operator follows from the
Riesz representation theorem.
[; ]
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
The following properties of the Hermitian adjoint of
bounded operators are immediate:
#
Involutivity:
# If is invertible, then so is , with
#
Anti-linearity:
#*
#* , where denotes the
complex conjugate of the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
# "
Anti-distributivity":
If we define the
operator norm of by
:
then
:
Moreover,
:
One says that a norm that satisfies this condition behaves like a "largest value", extrapolating from the case of self-adjoint operators.
The set of bounded linear operators on a complex Hilbert space together with the adjoint operation and the operator norm form the prototype of a
C*-algebra
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuous ...
.
Adjoint of densely defined unbounded operators between Hilbert spaces
Definition
Let the inner product
be linear in the ''first'' argument. A
densely defined operator from a complex Hilbert space to itself is a linear operator whose domain is a dense
linear subspace
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, li ...
of and whose values lie in . By definition, the domain of its adjoint is the set of all for which there is a satisfying
:
Owing to the density of
and
Riesz representation theorem,
is uniquely defined, and, by definition,
Properties 1.–5. hold with appropriate clauses about
domains and
codomains. For instance, the last property now states that is an extension of if , and are densely defined operators.
ker A=(im A)
For every
the linear functional
is identically zero, and hence
Conversely, the assumption that
causes the functional
to be identically zero. Since the functional is obviously bounded, the definition of
assures that
The fact that, for every
shows that
given that
is dense.
This property shows that
is a topologically closed subspace even when
is not.
Geometric interpretation
If
and
are Hilbert spaces, then
is a Hilbert space with the inner product
:
where
and
Let
be the
symplectic mapping, i.e.
Then the graph
:
of
is the
orthogonal complement of
:
The assertion follows from the equivalences
:
and
:
Corollaries
=A is closed
=
An operator
is ''closed'' if the graph
is topologically closed in
The graph
of the adjoint operator
is the orthogonal complement of a subspace, and therefore is closed.
=A is densely defined ⇔ A is closable
=
An operator
is ''closable'' if the topological closure
of the graph
is the graph of a function. Since
is a (closed) linear subspace, the word "function" may be replaced with "linear operator". For the same reason,
is closable if and only if
unless
The adjoint
is densely defined if and only if
is closable. This follows from the fact that, for every
:
which, in turn, is proven through the following chain of equivalencies:
:
=A = A
=
The ''closure''
of an operator
is the operator whose graph is
if this graph represents a function. As above, the word "function" may be replaced with "operator". Furthermore,
meaning that
To prove this, observe that
i.e.
for every
Indeed,
:
In particular, for every
and every subspace
if and only if
Thus,
and
Substituting
obtain
=A = (A)
=
For a closable operator
meaning that
Indeed,
:
Counterexample where the adjoint is not densely defined
Let
where
is the linear measure. Select a measurable, bounded, non-identically zero function
and pick
Define
:
It follows that
The subspace
contains all the
functions with compact support. Since
is densely defined. For every
and
:
Thus,
The definition of adjoint operator requires that
Since
this is only possible if
For this reason,
Hence,
is not densely defined and is identically zero on
As a result,
is not closable and has no second adjoint
Hermitian operators
A
bounded operator is called Hermitian or Self-adjoint operator">self-adjoint
In mathematics, and more specifically in abstract algebra, an element ''x'' of a *-algebra is self-adjoint if x^*=x. A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold.
A collection ''C'' of elements of a sta ...
In some sense, these operators play the role of the real numbers (being equal to their own "complex conjugate") and form a real vector space. They serve as the model of real-valued observables in
. See the article on
s for a full treatment.
the definition of adjoint needs to be adjusted in order to compensate for the complex conjugation. An adjoint operator of the antilinear operator on a complex Hilbert space is an antilinear operator with the property:
:
, and this is where adjoint functors got their name from.