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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 A is also sometimes called the Hermitian conjugate (after Charles Hermite) of A and is denoted by A* or A† (the latter especially when used in conjunction with the bra–ket notation).
Definition for bounded operators
Using the Riesz 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.
For unbounded operators
Following properties of Hermitian adjoint on bounded operators are immediate:
- A** = A – involutiveness
- If A is invertible, then so is A*, with (A*)−1 = (A−1)*
- (A + B)* = A* + B*
- (λA)* = , where A* denotes the complex conjugate of the complex number λ – antilinearity (together with 3.)
- (AB)* = B* A*
If we define the operator norm of A by
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 Hilbert space H together with the adjoint operation and the operator norm form the prototype of a C*-algebra.
Hermitian adjoint is defined for any densely defined operator, not necessarily a bounded one. Properties 1.–5. hold with appropriate clauses about domains and codomains. The relationship between the image of A and the kernel of its adjoint is given by:
- (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 A on a Hilbert space H is an antilinear operator A* : H → H with the property:
- Mathematical concepts
- Physical applications