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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 ...
A on a Euclidean vector space defines a Hermitian adjoint (or adjoint) operator A^* on that space according to the rule :\langle Ax,y \rangle = \langle x,A^*y \rangle, where \langle \cdot,\cdot \rangle 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 H. 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—H.


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 ...
A: H_1\to H_2 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 A^* : H_2 \to H_1 fulfilling :\left\langle A h_1, h_2 \right\rangle_ = \left\langle h_1, A^* h_2 \right\rangle_, where \langle\cdot, \cdot \rangle_ 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 H_i, which is linear in the first coordinate and antilinear in the second coordinate. Note the special case where both Hilbert spaces are identical and A 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 A: E \to F, where E, F are Banach spaces with corresponding norms \, \cdot\, _E, \, \cdot\, _F. Here (again not considering any technicalities), its adjoint operator is defined as A^*: F^* \to E^* with :A^*f = f \circ A : u \mapsto f(Au), I.e., \left(A^*f\right)(u) = f(Au) for f \in F^*, u \in E. 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 A: H \to E, where H is a Hilbert space and E is a Banach space. The dual is then defined as A^*: E^* \to H with A^*f = h_f such that :\langle h_f, h\rangle_H = f(Ah).


Definition for unbounded operators between Banach spaces

Let \left(E, \, \cdot\, _E\right), \left(F, \, \cdot\, _F\right) 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 A: D(A) \to F and D(A) \subset E, and suppose that A 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., D(A) is dense in E). Then its adjoint operator A^* is defined as follows. The domain is :D\left(A^*\right) := \left\. Now for arbitrary but fixed g \in D(A^*) we set f: D(A) \to \R with f(u) = g(Au). By choice of g and definition of D(A^*), f is (uniformly) continuous on D(A) as , f(u), = , g(Au), \leq c\cdot \, u\, _E. Then by Hahn–Banach theorem or alternatively through extension by continuity this yields an extension of f, called \hat defined on all of E. Note that this technicality is necessary to later obtain A^* as an operator D\left(A^*\right) \to E^* instead of D\left(A^*\right) \to (D(A))^*. Remark also that this does not mean that A can be extended on all of E but the extension only worked for specific elements g \in D\left(A^*\right). Now we can define the adjoint of A as :\begin A^*: F^* \supset D(A^*) &\to E^* \\ g &\mapsto A^*g = \hat f \end The fundamental defining identity is thus :g(Au) = \left(A^* g\right)(u) for u \in D(A).


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 ...
\langle\cdot,\cdot\rangle. 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 : \langle Ax , y \rangle = \left\langle x , A^* y\right\rangle \quad \mbox x, y \in H. 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 \left(A^*\right)^ = \left(A^\right)^* # 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 :\, A \, _\text := \sup \left\ then :\left\, A^* \right\, _\text = \, A\, _\text. Moreover, :\left\, A^* A \right\, _\text = \, A\, _\text^2. 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 \langle \cdot, \cdot \rangle 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 : \langle Ax , y \rangle = \langle x , z \rangle \quad \mbox x \in D(A). Owing to the density of D(A) and Riesz representation theorem, z is uniquely defined, and, by definition, A^*y=z. 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 y \in \ker A^*, the linear functional x \mapsto \langle Ax,y \rangle = \langle x,A^*y\rangle is identically zero, and hence y \in (\operatorname A)^\perp. Conversely, the assumption that y \in (\operatorname A)^\perp causes the functional x \mapsto \langle Ax,y \rangle to be identically zero. Since the functional is obviously bounded, the definition of A^* assures that y \in D(A^*). The fact that, for every x \in D(A), \langle Ax,y \rangle = \langle x,A^*y\rangle = 0 shows that A^* y \in D(A)^\perp =\overline^\perp = \, given that D(A) is dense. This property shows that \operatornameA^* is a topologically closed subspace even when D(A^*) is not.


Geometric interpretation

If H_1 and H_2 are Hilbert spaces, then H_1 \oplus H_2 is a Hilbert space with the inner product :\bigl \langle (a,b),(c,d) \bigr \rangle_ \stackrel \langle a,c \rangle_ + \langle b,d \rangle_, where a,c \in H_1 and b,d \in H_2. Let J\colon H\oplus H \to H \oplus H be the symplectic mapping, i.e. J(\xi, \eta) = (-\eta, \xi). Then the graph :G(A^*) =\ \subseteq H \oplus H of A^* is the orthogonal complement of JG(A): :G(A^*) = (JG(A))^\perp = \. The assertion follows from the equivalences : \bigl \langle (x, y) , (-A\xi, \xi) \bigr \rangle = 0 \quad \Leftrightarrow \quad \langle A\xi, x \rangle = \langle \xi, y \rangle, and :\Bigl \forall \xi \in D(A)\ \ \langle A\xi, x \rangle = \langle \xi, y \rangle \Bigr \quad \Leftrightarrow \quad x \in D(A^*)\ \&\ y = A^*x.


Corollaries


=A is closed

= An operator A is ''closed'' if the graph G(A) is topologically closed in H \oplus H. The graph G(A^*) of the adjoint operator A^* is the orthogonal complement of a subspace, and therefore is closed.


=A is densely defined ⇔ A is closable

= An operator A is ''closable'' if the topological closure G^\text(A) \subseteq H \oplus H of the graph G(A) is the graph of a function. Since G^\text(A) is a (closed) linear subspace, the word "function" may be replaced with "linear operator". For the same reason, A is closable if and only if (0,v) \notin G^\text(A) unless v=0. The adjoint A^* is densely defined if and only if A is closable. This follows from the fact that, for every v \in H, :v \in D(A^*)^\perp\ \Leftrightarrow\ (0,v) \in G^\text(A), which, in turn, is proven through the following chain of equivalencies: : \begin v \in D(A^*)^\perp &\Longleftrightarrow (v,0) \in G(A^*)^\perp \Longleftrightarrow (v,0) \in (JG(A))^\text = JG^\text(A) \\ &\Longleftrightarrow (0,-v) = J^(v,0) \in G^\text(A) \\ &\Longleftrightarrow (0,v) \in G^\text(A). \end


=A = A

= The ''closure'' A^\text of an operator A is the operator whose graph is G^\text(A) if this graph represents a function. As above, the word "function" may be replaced with "operator". Furthermore, A^ = A^, meaning that G(A^) = G^(A). To prove this, observe that J^* = -J, i.e. \langle Jx,y\rangle_ = -\langle x,Jy\rangle_, for every x,y \in H \oplus H. Indeed, : \begin \langle J(x_1,x_2),(y_1,y_2)\rangle_ &= \langle (-x_2,x_1),(y_1,y_2)\rangle_ = \langle -x_2,y_1\rangle_H + \langle x_1,y_2 \rangle_H \\ &= \langle x_1,y_2 \rangle_H + \langle x_2,-y_1 \rangle_H = \langle (x_1,x_2),-J(y_1,y_2)\rangle_. \end In particular, for every y \in H \oplus H and every subspace V \subseteq H \oplus H, y \in (JV)^\perp if and only if Jy \in V^\perp. Thus, J JV)^\perp= V^\perp and [(JV)^\perp^\perp = V^\text. Substituting V = G(A), obtain G^\text(A) = G(A^).


=A = (A)

= For a closable operator A, A^* = \left(A^\text\right)^*, meaning that G(A^*) = G\left(\left(A^\text\right)^*\right). Indeed, : G\left(\left(A^\text\right)^*\right) = \left(JG^\text(A)\right)^\perp = \left(\left(JG(A)\right)^\text\right)^\perp = (JG(A))^\perp = G(A^*).


Counterexample where the adjoint is not densely defined

Let H=L^2(\mathbb,l), where l is the linear measure. Select a measurable, bounded, non-identically zero function f \notin L^2, and pick \varphi_0 \in L^2 \setminus \. Define :A \varphi = \langle f,\varphi\rangle \varphi_0. It follows that D(A) = \. The subspace D(A) contains all the L^2 functions with compact support. Since \mathbf_ \cdot \varphi\ \stackrel\ \varphi, A is densely defined. For every \varphi \in D(A) and \psi \in D(A^*), :\langle \varphi, A^*\psi \rangle = \langle A\varphi, \psi \rangle = \langle \langle f,\varphi \rangle\varphi_0, \psi \rangle = \langle f,\varphi \rangle\cdot \langle \varphi_0, \psi \rangle = \langle \varphi, \langle \varphi_0, \psi \rangle f\rangle. Thus, A^* \psi = \langle \varphi_0, \psi \rangle f. The definition of adjoint operator requires that \mathopA^* \subseteq H=L^2. Since f \notin L^2, this is only possible if \langle \varphi_0, \psi \rangle= 0. For this reason, D(A^*) = \^\perp. Hence, A^* is not densely defined and is identically zero on D(A^*). As a result, A is not closable and has no second adjoint A^.


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 ...
if :A = A^* which is equivalent to :\langle Ax , y \rangle = \langle x , A y \rangle \mbox x, y \in H.; 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
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, ...
. See the article on self-adjoint operators for a full treatment.


Adjoints of antilinear operators

For an antilinear operator 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: : \langle Ax , y \rangle = \overline \quad \text x, y \in H.


Other adjoints

The equation : \langle Ax , y \rangle = \left\langle x, A^* y \right\rangle is formally similar to the defining properties of pairs of adjoint functors in
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, and this is where adjoint functors got their name from.


See also

* Mathematical concepts **
Hermitian operator In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to itse ...
**
Norm (mathematics) In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is ze ...
** Transpose of linear maps ** Conjugate transpose * Physical applications **
Operator (physics) In physics, an operator is a function over a space of physical states onto another space of physical states. The simplest example of the utility of operators is the study of symmetry (which makes the concept of a group useful in this context). Bec ...
**
†-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 ...


References

* . * . * {{DEFAULTSORT:Hermitian Adjoint Operator theory