In mathematics, specifically in functional analysis, each bounded 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 Template:Mvar may also be called the Hermitian adjoint, Hermitian conjugate or Hermitian transpose (after Charles Hermite) of Template:Mvar and is denoted by A* or A† (the latter especially when used in conjunction with the bra–ket notation).
Definition for bounded operators
Suppose Template:Mvar is a Hilbert space, with inner product . Consider a continuous linear operator A : H → H (for linear operators, continuity is equivalent to being a bounded operator). Then the adjoint of Template:Mvar is the continuous linear operator A* : H → H satisfying
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.
- A** = A – involutiveness
- If Template:Mvar is invertible, then so is A*, with (A*)−1 = (A−1)*
- (A + B)* = A* + B*
- (λA)* = Template:OverlineA*, where Template:Overline denotes the complex conjugate of the complex number λ – antilinearity (together with 3.)
- (AB)* = B* A*
One says that a norm that satisfies this condition behaves like a "largest value", extrapolating from the case of self-adjoint operators.
Adjoint of densely defined operators
A densely defined operator Template:Mvar on a Hilbert space Template:Mvar is a linear operator whose domain D(A) is a dense linear subspace of Template:Mvar and whose co-domain is Template:Mvar. Its adjoint A* has as domain D(A*) the set of all y ∈ H for which there is a z ∈ H satisfying
Properties 1.–5. hold with appropriate clauses about domains and codomains. For instance, the last property now states that (AB)* is an extension of B*A* if Template:Mvar, Template:Mvar and Template:Mvar are densely defined operators.
- (see orthogonal complement)
Proof of the first equation:
which is equivalent to
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. 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 Template:Mvar on a Hilbert space Template:Mvar is an antilinear operator A* : H → H with the property:
- Mathematical concepts
- Physical applications