Projection-valued measure

In mathematics, especially in the functional analysis, a spectral measure is a mapping that certain subsets of a fixed amount selected allocates orthogonal projections of a Hilbert space. Spectral measures are used to formulate results in the spectral theory of linear operators, such as the spectral theorem for normal operators. Besides the term, but with a different meaning, used in the stochastics.

Definition

Let be a measurable space, a real or complex Hilbert space, the Banach space of continuous linear operators on and the set of orthogonal projectors. A spectral measure for the triple is an illustration with the following properties:

The quadruple is called a Spektralmaßraum.

Frequently, the mapping is defined in this way as a partition of unity: means ( engl. resolution of the identity ). It is also common from a projector -valued measure (English: projection -valued measure, often short PVM ) to speak.

If a topological space, its topology and its Borel algebra, so is called a spectral measure, which the Borel space measurement is based on a Borel spectral measure. Is specially respectively, this means Borel spectral measure a real or complex spectral measure. The institution of a Borel Spektralmaßes than

Defined. This is the complement of the largest subset of the open is for.

Properties

It is a spectral measure for the date. Then the following statements hold:

• Modularity: It applies to everyone.
• Multipikativität: It applies to everyone. In particular, the projectors and commute with each other and the image of is perpendicular to the image of if and only if.

In particular, each spectral measure is a finitely additive measure vectorial.

Substituting for, then for all due to the polarization identity

In the complex case and

In the real case. In particular the dimensions are known, if the dimensions are known, so that they often work only with the latter.

Equivalent definition

Frequently one finds the following characterization of spectral masses in the literature as definition. A picture is exactly then a spectral measure if

The term partition of unity for can now be explained as follows. Is a countable decomposition of in measurable quantities, shall apply

Or

The orthogonal sum in the sense of Hilbert spaces of the family of closed subspaces is. This corresponds to the fact that form the eigenspaces of a normal operator of an orthogonal sum decomposition of.

Examples

Whether it is a normal linear operator. Then, the spectrum of non-empty and is composed of the eigenvalues ​​of. The eigenspaces of the pairwise distinct eigenvalues ​​of are orthogonal and have as (internal ) direct sum. This is equivalent to that

Applies. The orthogonal projection of on is the eigenspace of the eigenvalue. This representation is obtained from the

" Spectral resolution " of the spectral measure is of

Is any normal operator, then the spectrum of can be continuous or pile up at one point and substituting above sum by a continuous summation term, namely a ( operatorwertiges ) integral.

• Every normal operator of a Hilbert space defines a spectral measure. According to the spectral theorem for normal operators, the operator is uniquely described by this spectral measure.

• Let L2 [ 0,1], the Hilbert space of square - summable in the sense of Lebesgue functions on the unit interval and the Borel algebra. For a much more limited function to denote the induced multiplication operator. Identifies the characteristic function of a Borel set of the unit interval, and identifying, then this is a spectral measure for the tuple is defined. This is the spectral measure of the multiplication operator.

Integration with respect to a Spektralmaßes

It is a Spektralmaßraum. Using the complex to associate degree can be for certain - measurable functions a (usually unbounded ) linear operator

Explain the Hilbert space. This operator is called Spektralintegral of and the process by which it arises from when integrating with respect to the Spektralmaßes.

Spectral measure of a normal operator

Let be a Hilbert space and a normal operator with spectrum. Then you explain as follows a spectral measure on the Borel algebra. It is the functional calculus of functions of bounded Borel. Since a morphism of algebras is given for each Borel set of the spectrum of by an orthogonal projection of. It can be shown that a spectral measure is the spectral measure of the normal operator. The spectral theorem for normal operators now states that

Applies. The focus is on the right side of this equation the Spektralintegral the limited Borel function with respect to the Spektralmaßes.

Spectral function

Definition of the spectral function

A family of orthogonal projectors is called a Spektralfamilie or spectral function if the following conditions are met:

• .
• .
• The family is right-continuous, that is true in the sense.
• The family is monotone increasing: Applies shall prevail. This condition is equivalent to the following condition: For all.

Here are all occurring limits in the sense of the strong operator topology, ie to consider pointwise.

Relationship with the spectral measure

The concept of Spektralfamilie was historically preceded the concept of Spektralmaßes and was introduced by John von Neumann called partition of unity. The relationship between the two terms is given as follows: For every real spectral measure exactly heard a spectral function and vice versa. In this case, the spectral measure and the spectral function mutually determined by the relationship

The support of the spectral function is the set of

Using a spectral function whose support is compact, can be a based on the Stieltjes integral for a continuous function, as

Listed, define operator. This is clearly determined by the relationship that he

Satisfied, where now the right-hand a conventional Stieltjes integral stands. It is then

When the corresponding spectral measure referred to.

Spectral measure of a bounded self-adjoint operator

The spectral function of a bounded self-adjoint operator has compact support in, where

Or

Be. sometimes referred to as Spektralprojektion. The image of this orthogonal projection It is envisioned as a kind of generalized eigenspace.

Spectral measure unbounded self-adjoint operators ( quantum mechanics)

The measurable quantities of quantum mechanics correspond ( almost entirely unrestricted, densely defined ) is significantly self-adjoint Hilbert space operators on separable Hilbert spaces ( " observables " → Mathematical structure of quantum mechanics ), with a spectral decomposition into three parts, in accordance with the above statements:

All observables show such a division, and have conventional spectral measures and usual spectral projections. However, the above compactness of the spectrum does not apply.

The division into three parts for a total of at weighting the squares of the contributions of the eigenfunctions and the generalized eigenfunctions, the exact value of 1, consistent with the probabilistic interpretation of quantum mechanics.

In the case of a pure point spectrum, the spectral characteristics correspond to the postulate of the completeness of the eigenfunctions ( development set). In the case of an additional absolutely - continuous spectral physicists work, as mentioned, with so-called generalized eigenfunctions and wave packets ( the context of the spectral measure is derived from the distribution theory of so-called Gelfandsche Raumtripel ). A singular continuous spectral component is usually not discussed, other than, for example, in crystals with a special " incommensurable " magnetic fields. Details in textbooks of quantum mechanics and the measure theory of real functions.