In mathematics, specifically in functional analysis, each bounded linear operator on a complex Hilbert space has a corresponding Hermitian adjoint. Adjoints of operators generalize conjugate transposes of square matrices to infinite-dimensional situations. If one thinks of operators on a complex Hilbert space as generalized complex numbers, then the adjoint of an operator plays the role of the complex conjugate of a complex number. In a similar sense, one can define an adjoint operator for linear operators between Banach spaces. The adjoint of an operator may also be called the Hermitian conjugate, Hermitian or Hermitian transpose of and is denoted by or . Confusingly, may also be used to represent the conjugate of.
Informal definition
Consider a linear operator between Hilbert spaces. Without taking care of any details, the adjoint operator is the linear operator fulfilling where is the inner product in the Hilbert space. Note the special case where both Hilbert spaces are identical and is an operator on that Hilbert space. When one trades the dual pairing for the inner product, one can define the adjoint of an operator, where are Banach spaces with corresponding norms. Here, 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 normed spaces
Let be Banach spaces. Suppose is a linear operator which is densely defined. 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 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
Suppose is a complex Hilbert space, with inner product. Consider a continuous linear 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
"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.
Adjoint of densely defined unbounded operators between Hilbert spaces
A densely defined operator from a complex Hilbert space to itself is a linear operator whose domain is a dense linear subspace of and whose values lie in. By definition, the domain of its adjoint is the set of all for which there is a satisfying and is defined to be the thus found. 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. The relationship between the image of and the kernel of its adjoint is given by: These statements are equivalent. See orthogonal complement for the proof of this and for the definition of. Proof of the first equation: The second equation follows from the first by taking the orthogonal complement on both sides. Note that in general, the image need not be closed, but the kernel of a continuous operator always is.
A bounded operator is called Hermitian or self-adjoint if which is equivalent to In some sense, these operators play the role of the real numbers 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 on a complex Hilbert space is an antilinear operator with the property:
Other adjoints
The equation is formally similar to the defining properties of pairs of adjoint functors in category theory, and this is where adjoint functors got their name from.
