In mathematics, specifically in functional analysis, each bounded linear operator on a complex 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 complex 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 may also be called the Hermitian adjoint, Hermitian conjugate or Hermitian transpose (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
Suppose H is a complex 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 A 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 A is invertible, then so is A*, with (A*)−1 = (A−1)*
- (A + B)* = A* + B*
- (λA)* = λA*, where λ 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 complex Hilbert space H together with the adjoint operation and the operator norm form the prototype of a C*-algebra.
Adjoint of densely defined operators
A densely defined operator A from a complex Hilbert space H to itself is a linear operator whose domain D(A) is a dense linear subspace of H and whose values lies in H. By definition, the domain D(A*) of its adjoint A* is the set of all y ∈ H for which there is a z ∈ H satisfying
and A*(y) is defined to be the z thus found.
Properties 1.–5. hold with appropriate clauses about domains and codomains.[clarification needed] For instance, the last property now states that (AB)* is an extension of B*A* if A, B and AB are densely defined operators.
The relationship between the image of A 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 .
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.[clarification needed]
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 complex Hilbert space H is an antilinear operator A* : H → H with the property:
- Mathematical concepts
- Physical applications
- David A. B. Miller (2008). Quantum Mechanics for Scientists and Engineers. Cambridge University Press. pp. 262, 280.
- Reed & Simon 2003, pp. 186–187; Rudin 1991, §12.9
- See unbounded operator for details.
- Reed & Simon 2003, pp. 252; Rudin 1991, §13.1
- Rudin 1991, Thm 13.2
- See Rudin 1991, Thm 12.10 for the case of bounded operators
- The same as a bounded operator.
- Reed & Simon 2003, pp. 187; Rudin 1991, §12.11
- Reed, Michael; Simon, Barry (2003), Functional Analysis, Elsevier, ISBN 981-4141-65-8.
- Rudin, Walter (1991), Functional Analysis (second ed.), McGraw-Hill, ISBN 0-07-054236-8.