Spectral theorem

The term spectral theorem refers to several inter- related mathematical statements from linear algebra and functional analysis. The simplest way of making a statement about the diagonalizability a certain class of matrices. The other consideration here Spektralsätze apply this principle to operators between infinite-dimensional spaces. The name derives from the " spectrum" of the eigenvalues.

• 2.1 statement
• 2.2 projection version of the Spectral Theorem

• 3.1 statement
• 3.2 related to the previous Spektralsätzen
• 3.3 Example
• 3.4 Measurable functional calculus

Spectral theorem for endomorphisms of finite dimensional vector spaces

Statement

Exists for an endomorphism of a finite-dimensional unitary vector space ( or ) if an orthonormal basis of eigenvectors, when he is normal and to be among all eigenvalues ​​.

In matrix manner of speaking, this means that a matrix is unitarily diagonalizable if and only if it is normal and has only eigenvalues ​​. Another common formulation is that a matrix is exactly then normal, if it is unitarily diagonalizable, ie a unitary matrix (same dimension) exists so that

With, a diagonal matrix with the eigenvalues ​​of on the main diagonal, is.

Comments

• For the condition that all eigenvalues ​​lie in, always satisfied ( is algebraically closed by the fundamental theorem of algebra), so here all normal matrices are diagonalizable. For this is not true.

• A self-adjoint endomorphism or a Hermitian matrix has only real eigenvalues. The spectral theorem therefore states that all Hermitian matrices are diagonalizable and an endomorphism exactly is self-adjoint if there is an orthonormal basis of eigenvectors and eigenvalues ​​are real.

Spectral theorem for compact operators

Statement

Be a Hilbert space and a compact linear operator, which is normally self-adjoint in the case or if. Then there exists a (possibly finite ) orthonormal system and a null sequence such that

As well as

Applies to all. These are for all eigenvalues ​​of and is an eigenfunction. Also, applies where the operator norm is.

Projection version of the Spectral Theorem

One can reformulate the spectral theorem for compact operators with the help of orthogonal projections. Be again a Hilbert space and a compact linear operator, which in the case of normal or self-adjoint in the case ist.Mit the Orthgonalprojektion referred to belonging to eigenspace. Thus, the operator has the representation, the dimension of the eigenspace and the standard basis of the eigenspace is. Then we can reformulate the spectral theorem to that there exists a null sequence of eigenvalues ​​, so that

Applies to all. This series converges not only pointwise but also with respect to the operator norm.

Spectral theorem for bounded operators

Statement

Be a Hilbert space and a self-adjoint continuous linear operator. Then a uniquely determined spectral measure with compact support exists in with

The Borel σ algebra referred to by the amount of bounded operators and the range of.

Related to previous Spektralsätzen

• Is finite, that is true, then the self-adjoint operator has the pairwise distinct eigenvalues ​​and we have already shown in the article the orthogonal projection on the eigenspace of. The spectral measure of is then for all given. Therefore, reducing the spectral theorem for bounded operators with the spectral theorem from linear algebra.
• Be a linear compact operator, as was also shown in the article that there is a spectral theorem for such operators. Be the sequence of eigenvalues ​​of and is chosen again as a spectral measure, the sum then in general countably many summands and has pointwise, but not with respect to the operator norm converges, then the spectral theorem for bounded operators simplifies to Therefore, the spectral theorem for bounded operators also includes the spectral theorem for compact operators.

Example

The operator defined by is self-adjoint and has no eigenvalues. The spectral measure with a spectral measure with compact support. It represents, since it is

Measurable functional calculus

Be a self-adjoint operator. The measurable functional calculus is a uniquely determined, steady, involutive algebra homomorphism. Using the spectral decomposition to obtain a simple illustration of this figure. It is namely

Spectral theorem for unbounded operators

Is a densely defined normal operator on a complex Hilbert space, then there exists a uniquely determined spectral measure on the Borel sets of such that the following holds (whether the spectrum of ):

• For with a lot.
• With For an open set.

A self-adjoint operator is normal with real spectrum; so you can restrict to real numbers, the above integral.

The domain is given by

And the quadratic form field by

The latter is obviously the maximal domain of definition for the associated quadratic form which is particularly important in quantum mechanics.

An equivalent formulation of the Spectral is that unitarily equivalent to a multiplication operator via a space ( a measure space ) with a measurable complex function; is self-adjoint, so is real-valued.

A normal operator in the complex can be generally written as a sum of two with the real or the imaginary unit multiplied, each of commuting self-adjoint operators ( " real part " i " imaginary " ), following also applies - because of the interchangeability of the - that the operator and the operator have the same eigenvectors (despite possibly different eigenvalues ​​). For example, a function of the self-adjoint operator to be removed with suitable f2. Then it would ultimately only a single ( reelle! ) spectral representation of, for example, the of, and it would apply, for example, that   and is.

Role in quantum mechanics

A special role is played by the spectral theorem ( " development kit " ) in quantum mechanics because

• First, the measurable variables ( " observables " ) by the self-adjoint operators are shown in the Hilbert space,
• Second, the possible values ​​are determined by the spectral values ​​( more precisely, in the case of a so-called point of the spectrum ( " discrete spectrum " ) by the so-called eigen values ​​, in the case of a so-called " continuous spectral " by the respective spectral intervals), and
• Thirdly, to an eigenvalue ( point spectrum ) are given corresponding measurement probabilities in the given quantum state by the squared modulus of the scalar products of the state function and the eigenfunctions of the observable ( in the case of a continuous spectral component can thus approximate ).

History

The spectral theorem for compact self-adjoint operators and self-adjoint bounded operators for particular go back to work of David Hilbert. He published in 1906 in his fourth release a proof of these statements. Hilbert representation of sets of course very different from today's presentation. Instead of Spektralmaßes he used the Stieltjes integral, which Jean Thomas Stieltjes had until 1894 introduced for the study of continued fractions. According to Hilbert, among other evidence of Riesz (1930-1932) and Lengyel and Stone (1936) and for the unrestricted case of Leinfelder (1979 ) were found for the spectral theorem for bounded and unbounded operators.