Trace class

The trace class operators are studied in the mathematical discipline of functional analysis. They form an important class of linear operators on infinite-dimensional spaces. For them, as opposed to general operators, some properties of the finite-dimensional case are preserved, especially affecting its representation as a sum of one-dimensional operators. In important cases, the known from linear algebra concept of track transfers on these operators, which has led to their name. In quantum mechanics, the trace class operators occur as the density matrix.

Alexander Grothendieck met with his investigation of the sentence from the core of the distribution theory also to operators of the kind discussed here and named it nuclear operators (Latin = nucleus core). This then led to the concept of the nuclear space.

This article addresses the first nuclear operators on Hilbert spaces, then generalized to Banach spaces and finally on locally convex spaces.

Motivation

Be a vector space over the field of real or complex numbers. A one-dimensional operator is an operator of the mold and wherein the dual space of designated. In linear algebra, i.e. in the case where each linear map can be represented as a matrix with respect to a base. For then applies

.

A is thus a sum of one-dimensional operators. In order to apply this to infinite-dimensional spaces, one has to form infinite sums of one-dimensional operators, and therefore make arrangements for their convergence. This leads to the following definition:

Definition

Let and be two normed vector spaces. An operator is called nuclear if there are consequences in and with

And

For everyone. Such a formula is called for a nuclear representation of. However, this is not unique.

The nuclear norm or trace norm of a nuclear operator is defined as

Where the infimum is formed over the consequences in and which result in a nuclear representation of.

Examples

• Let and be defined by. Then with nuclear. In the Hilbert space case equality holds.
• Be steady, be defined by. Then with nuclear.
• Be defined by. Then is a compact operator which is not nuclear.

Simple properties

Let the set of all nuclear operators. Is complete, so is a Banach space with the nuclear norm. The operators with nite dimensional lie close in and each nuclear operator is compact.

The nuclear operators have the so-called ideal property: Let and be normed spaces, and is nuclear and are continuous linear operators. Then also nuclear and it is, the operator norm is. It is always

Specifically, an ideal in the algebra of continuous linear operators on, and with the nuclear norm is a Banach algebra.

Nuclear operators on Hilbert spaces

In Hilbert space the situation is particularly simple. In these rooms, the nuclear operators have been first studied in 1946 by Robert shade and John von Neumann.

Each is by the theorem of Riesz - Fréchet of the mold with a. A representation of a nuclear operator therefore has the form

With and

Is an arbitrary orthonormal basis of so converges for each

Where the left sum as the limit of the net of all finite partial sums in reading (that is as unconditional convergence). This number is independent of the choice of orthonormal basis and also independent of the choice of nuclear representation, it is called the trace of and is denoted by. Because the English word for trace trace it often finds the designation.

Is self-adjoint and is the result of counted with multiplicities of the eigenvalues ​​of, and so applies. For the general eigenvalue sequence is absolutely summable and it is.

As a further characterization, one can show that an operator is exactly then nuclear, if it is the product of two Hilbert - Schmidt operators.

Plays a central role in the duality theory of operator algebras. Denote the algebra of compact linear operators. Each defined on by a steady, linear functional. One can show that an isometric isomorphism, which is provided with the nuclear norm and the operator norm. In this sense applies. Similarly, each defined by the formula a continuous linear functional on and one can show that an isometric isomorphism is when you know with the nuclear norm and the operator norm again. In this sense applies. In particular, is that is to say the rooms and are not reflexive in unendlichdimensionalem Hilbert space.

An analogy to sequence spaces

The following table provides an analogy between sequence spaces of complex numbers and operator algebras on a Hilbert space. For the purposes of this analogy, one can consider the nuclear operators as a non- commutative version of the sequences, it is at least a hint.

Nuclear operators on Banach spaces

The investigation of nuclear operators on Banach spaces began in 1951 with a thesis of AF Ruston. Because of the lack here orthonormal bases, the situation is not as simple as in the Hilbert space case, also significantly different methods are required.

While the eigenvalue sequence of a nuclear operator with respect to the above, is absolutely summable in the Hilbert space case, one can only prove the following weaker statement in the Banach space case:

If a Banach space and is the eigenvalue sequence of a nuclear operator, it shall and.

This result can not improve; RJ Kaiser and Ronald James Retherford have a nuclear operator from specified for a given sequence with this eigenvalue sequence. By a theorem of Johnson, King, Maurray and Retherford is a Banach space is isomorphic to a Hilbert space when the eigenvalue result of each nuclear operator is made ​​.

The trace of a nuclear operator can not be defined for all Banach spaces. Is a representation of a nuclear operator from given, the Hilbert space case suggests the definition. This number turns out to be exactly well-defined, that is, as independent of the selected nuclear representation if the Banach space has the approximation property.

The in the Hilbert space case this duality generalizes as follows Banach spaces with approximation property. Each defined on a continuous linear functional, wherein when a nuclear representation of is. The approximation property ensures the well- definedness, ie the independence of the choice of nuclear representation. One can show that an isometric isomorphism is when you know with the nuclear norm and the operator norm. In this sense. If, therefore, in addition reflexive, one has as in the Hilbert space case.

Nuclear operators on locally convex spaces

Alexander Grothendieck in 1951 began with the investigation of nuclear operators between locally convex spaces. Since one on locally convex spaces has no standard is available, the definition must be formulated as follows: A linear operator is called nuclear if there exists a representation of the type

Are, with

• ,
• An equicontinuous sequence in the strong dual space (ie there exists a continuous seminorm on with for all)
• A bounded sequence in is.

Since the required uniform continuity is equivalent to the Banach space case the boundedness, the definition given here results in Banach space case on the same concept of the nuclear operator as defined above.

The ideal property generalized to locally convex spaces: Is nuclear and are continuous and linear operators between locally convex spaces, so is nuclear. Nuclear operators are continuous and, if complete, even compact. One can show that for every nuclear operator itself a nuclear operator between normed spaces and continuous linear operators are with. This can be attributed to the study of nuclear operators between locally convex spaces to the standardized case.

In the locally convex theory, the nuclear operators play an important role in the context of nuclear spaces.

Application in statistical physics

The physical area of statistical physics is based on the central assumption that the track of each with the exponential function of the so-called Hamiltonian ( energy operator ) at the temperature weighted measure of ( Observable ) exists quantum statistics, and that was despite the Hamiltonian itself by no means belongs to the trace class and usually also for the ( self-adjoint only! ) operator applies the same. Still applies to the thermal expectation value of the considered measure due to this assumption, the relationship

In other words, the bracketed expressions deal iW with nuclear spaces and the operators defined therein or measured variables.