Self-adjoint operator

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In mathematics, a self-adjoint operator is a densely defined linear operator mapping a complex Hilbert space onto itself and which is invariant under the unary operation of taking the adjoint. That is, if A is an operator with a domain which is a dense subspace of a complex Hilbert space H then it is self-adjoint if , where denotes the adjoint operator of A. Note that the adjoint of any densely defined linear operator is always well-defined (in fact, the denseness of the domain of an operator is necessary for the existence of its adjoint) and two operators A and B are said to be equal if they have a common domain and their values coincide on that domain.

On an infinite dimensional Hilbert space, a self-adjoint operator can be thought of as the analogy of a real symmetric matrix (i.e., a matrix which is its own transpose) or a Hermitian matrix in (i.e., a matrix which is its own Hermitian transpose) when these matrices are viewed as (bounded) linear operators on and , respectively.

Special properties of a self-adjoint operator

The self-adjointness of an operator entails that it has some special properties. Some of these properties include:

1. The eigenvalues of a self-adjoint operator are real. As a special well-known case, all eigenvalues of a real symmetric matrix and a complex Hermitian matrix are real.

2. By the von Neumann’s spectral theorem, any self-adjoint operator X (not necessarily bounded) can be represented as

where is the associated spectral measure of X (in particular, a spectral measure is a Hilbert space projection operator-valued measure)

3. By Stone’s Theorem, for any self-adjoint operator X the one parameter unitary group defined by , where is the spectral measure of X, satisfies:

for all u in the domain of X. One says that the operator -iX is the generator of the group U and writes: .

Examples of self-adjoint operators

As mentioned above, a simple instance of a self-adjoint operator is a Hermitian matrix.

For a more advanced example consider the complex Hilbert space of all complex-valued square integrable functions on with the complex inner product , and the dense subspace of of all infinitely differentiable complex-valued functions with compact support on . Define the operators Q, P on as:


where is the real valued Planck's constant. Then Q and P are self-adjoint operators satisfying the commutation relation on , where I denotes the identity operator. In quantum mechanics, the pair Q and P is known as the Schrödinger representation, on the Hilbert space , of canonical conjugate position and momentum operators q and p satisfying the canonical commutation relation (CCR) .

Further reading

  1. K. Yosida, Functional Analysis (6 ed.), ser. Classics in Mathematics, Berlin, Heidelberg, New York: Springer-Verlag, 1980.
  2. K. Parthasarathy, An Introduction to Quantum Stochastic Calculus, ser. Monographs in Mathematics, Basel, Boston, Berlin: Birkhauser Verlag, 1992.