Spectrum (functional analysis)

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The spectrum of a linear operator is a term from the functional analysis, a branch of mathematics. In the finite-dimensional linear algebra we consider endomorphisms, which are represented by matrices and their eigenvalues. The generalization to the infinite-dimensional is considered in the functional analysis. The spectrum of an operator can be thought of as a set of generalized eigenvalues. These are called spectral.

• 4.1 Examples 4.1.1 matrices
• 4.1.2 functions

Context of spectral theory to the eigenvalue theory

The spectral theory of linear operators from functional analysis is a generalization of the eigenvalue theory from linear algebra. In linear algebra endomorphisms are considered on finite dimensional vector spaces. The numbers for which the equation

Solutions, ie equal to the zero vector has, eigenvalues ​​are called, with a matrix representation of the selected endomorphism. Eigenvalues ​​are so figures for the inverse matrix does not exist in the unit, that is, the matrix is ​​not injective or onto. However, considering infinite-dimensional spaces it is necessary to distinguish whether the operator is invertible, and not injective or not surjective. In the infinite-dimensional case, it follows from the injectivity does not automatically surjectivity, as in the finite case. Hereinafter, the term spectrum is explained in the functional analysis.

Definition

The spectrum of an operator is the set of all elements of the number field (usually the complex numbers ) for which the difference of the operator with the imaging times of the identical

Is not limited invertible. The spectrum of the operator is referred to, and the elements of the spectrum, called spectral values.

The spectrum of linear operators

The above definition can be applied in different contexts. In this section the spectrum of linear operators of a vector space is considered. The spectral theory of linear operators can, however, only to a certain extent expand when the amount of considered operators is specified. For example, you might be restricted to limited, compact and self-adjoint operators. Below is a linear operator on a complex Banach space.

Resolvent

The resolvent set consists of all complex numbers, so that there is a limited to defined operator with

The operator is called the resolvent of the operator. The complement of the resolvent set is the set of complex numbers for which the resolvent does not exist or is unlimited, so the spectrum of the operator and is denoted by.

Distribution of the spectrum

The spectrum can be divided into different fractions. Once a sub-division into the discrete spectrum, the continuous spectrum, and the residual spectrum is made. These components of the spectrum to some extent differ in the reason for the non-existence of a bounded resolvent. A different separation of the spectrum is in the discrete and main spectrum. For the spectrum of a self-adjoint operator, there is the third possibility it into a point and to divide a continuous spectrum, it is described in the section on self-adjoint operators. Here, the continuous spectrum is not equivalent to a self-adjoint operator continuous spectrum, which is defined in the following subsection.

Point, steady - and residual spectrum

The point spectrum

If the operator is not injective, that is, it is an element of the dot range of. The elements of the point spectrum are called eigenvalues.

The continuous spectrum

However, if the operator is injective, not onto, but has a dense image, that is, there is an inverse, but which is defined on a thick part of the Banach space, then a member of the continuous spectrum of.

The residual spectrum

If the operator is injective, but has no dense in the Banach space image, then is an element of Residualspektrums of. In this case is not particularly onto. The inverse operator to exist, but only on a non- dense subspace of defined.

Discrete and essential spectrum

The set of all isolated spectral values ​​with finite multiplicity is called the discrete spectrum and recorded. The complement is called the essential spectrum of. However, there are other equivalent definitions to this definition of the essential and the discrete spectrum.

Example

An interesting example is the multiplication operator on a function space, which reflects the function of the function, ie, with.

• Looking on the space of bounded functions with the supremum norm, so its range is the interval and all spectral values ​​belong to the point spectrum.

• Looking at it on the Hilbert space of square integrable functions, the spectrum is again the interval and all spectral belong to the continuous spectrum.

• Finally Looking at him on the space of continuous functions, its spectrum is the interval and all spectral values ​​belong to the residual spectrum.

Properties of Compact Operators

The compact operators form a Banach space of bounded sets onto relatively compact sets of the same Banach space. This class of operators is itself a Banach algebra, which also forms a norm - closed ideal in the algebra of all bounded operators.

The spectrum of compact operators is surprisingly easy in the sense that it consists almost entirely of eigenvalues. This result goes back to Frigyes Riesz and is accurate:

For a compact operator on an infinite-dimensional Banach space that a spectral and each is an eigenvalue of finite multiplicity, that is the core of is finite, and does not have a different point of accumulation.

Properties of self-adjoint operators

The studied in quantum mechanics, operators are usually unlimited self-adjoint operators on Hilbert spaces. Statements about their spectrum are very essential. Only the spectral values ​​are measurable and it is the following statement.

The spectrum of a self-adjoint operator (or a so-called normal operator - which is simply the sum of two mutually commuting self-adjoint operators and, where is the imaginary unit ) can be decomposed into three parts:

Together, all three shares, with weighting function with the squares, exactly the value, according to the probabilistic interpretation of quantum mechanics.

Spectral theory for elements of a Banach algebra

Cross out the additional requirement of the boundedness of the inverse, so the above definition can be applied to elements of an operator algebra. Under an operator algebra is usually understood to be a Banach algebra with unit element and the inversion of elements results in this context makes sense only if the inverse is again an element of the algebra. Since such operators are not defined by their action on any vector space (that do not actually operate ), there is no a priori concept of boundedness of such operators. However, can these always as linear operators representing on a vector space, such as multiplication operators on the Banach algebra itself then these operators to bounded operators on a Banach space. In particular, the set of bounded operators is the standard example of an operator algebra. Also, the previously mentioned compact operators form an operator algebra. Therefore, the spectral theory for Banach algebras includes these two classes of linear operators.

Examples

Matrices

In linear algebra, the n × n matrices with complex entries form an algebra with respect to the usual addition and scalar multiplication ( component-wise ) and the matrix multiplication. The matrices can be viewed both as an example of actual operators in their capacity as linear mappings of, therefore, as well as an example of an operator algebra, but it is irrelevant in this context which operator norm is chosen for the matrices. Since all linear transformations of a finite dimensional space are automatically set to be limited, this term can be left in the definition here ignored.

A matrix can be inverted, if there is a matrix, so that ( unit matrix ) is. This is exactly the case when the determinant does not vanish. Therefore, a number is then a spectral value, if valid. However, since this is precisely the characteristic polynomial of the matrix in, is exactly then a spectral value when an eigenvalue of the matrix. In linear algebra, the spectrum of a matrix, therefore, denotes the set of eigenvalues.

Functions

The continuous functions on the interval of values ​​in the complex numbers form (eg as the supremum norm, but is not relevant here ) a Banach, the sum of two functions, and the product of two functions is defined pointwise:

A function is then called in this algebra is invertible if there is another function, so that (one function), that is if there is a function whose values ​​are precisely the reciprocals of. You can see now a fast, that a function is invertible if it does not have the function value and the inverse in this case the pointwise inverse function values ​​( reciprocals ) of the original function has:

A number is thus a spectral value if the function is not invertible, so has the function value. Of course this is exactly the case when a function value is. The spectrum of a function is therefore exactly her picture.

Properties

The spectral theory of the elements of Banach algebras with identity is an abstraction of the theory of bounded linear operators on a Banach space. The introductory examples are special cases of this theory, and in the first example, the norm of the functions under consideration has to be specified. If one chooses, for example, the Banach space of continuous functions on a compact space with the supremum norm, so, this example probably the most important case of an abelian Banach algebra with identity Represents the second example finds its place in this theory as a typical example of a nite nonabelian Banach algebra, a suitable standard is to choose the matrices. The spectrum of an operator was in the first case the value range and because the functions to be continuous on a compact set any compact subset of. In the second case, the spectrum is a finite set of points, and therefore also compact. This fact can be proved also in the abstract case:

From this set, the supremum follows immediately that there is a largest magnitude spectral value, because

Is assumed on the compact spectrum. We call this value the spectral radius of. In the example, the algebra of continuous functions one sees an immediate effect that the spectral radius corresponds exactly to the standard of the elements. From linear algebra, however, we know that this is not true for matrices in general, as for example, the matrix

Only the intrinsic value has, and therefore is, but the norm of the matrix ( no matter which ) is not. The spectral radius is generally actually smaller than the norm, but it is

Other applications

• The spectrum of the Hamiltonian in quantum mechanics are the possible energy values ​​that can be measured on the system under consideration. The Hamiltonian is therefore (and because it determines the dynamics of the system, see Mathematical Structure of Quantum Mechanics ) is a particularly important special case for the self-adjoint operators in general unbounded in the Hilbert space. This represents the quantum -mechanical states. The self-adjointness of the operator ensures, as already mentioned above that the possible values ​​( spectral ) are real numbers.
• In classical mechanics and statistical mechanics observables are modeled by functions on the phase space. Quite analogously to quantum mechanics also applies in this case that the possible values ​​are the spectral values ​​of the observables, so in this case simply the function values ​​.
• In algebraic quantum theory observables are introduced as abstract elements of certain C *-algebras (special Banach algebras ). Without a concrete representation of this algebra to be specified as set of linear operators on a Hilbert space, it allows the Spektralkalkül these algebras then to calculate the possible values ​​of the observables. The states of the physical system are then introduced not as vectors in Hilbert space, but as linear functionals on the algebra. The classical theories such as the classical (statistical ) mechanics, can be viewed as special cases in this picture, in which the C *-algebra is abelian.