Tensor product of Hilbert spaces

The considered in the mathematical branch of Functional Analysis formation of Hilbert space tensor is a method to compose new from Hilbert spaces Hilbert spaces. A purely algebraic form of the tensor product is not enough, that way you will not receive full spaces in general. Also examined in the Banach space injective and projective tensor products do not lead to the desired result, since one does not come in this way in general Hilbert spaces, that is, the standards are not defined by a scalar product.

Although scalar products in Hilbert spaces is not bilinear, but only sesquilinear, but nevertheless, it should be possible to continue them on algebraic tensor products of Hilbert spaces, since tensor products are indeed made ​​certain extent for bilinear mappings. They would then have at least a pre-Hilbert space, which you only have to complete in order to obtain a Hilbert space. This is precisely what proves to be successful. Below are just a complex Hilbert spaces are considered, which are important for many applications. The construction of tensor products of real spaces runs quite similar and in some details even easier.

Definition

Let and two - Hilbert spaces. The scalar products are always named, to specify the name of the Hilbert space is given if added as an index. Then one can show:

On the algebraic tensor product there is a unique sesquilinear form with the property

Completing the Prähilbertraums means Hilbert space and tensor and is designated by. Some authors use the algebraic tensor product and then write to complete, others use it for both, and indicate possible ambiguities or no use for the algebraic tensor product of a different notation, as done in this article.

Properties

• The Hilbert space tensor product is easy to finally extend many Hilbert spaces by induction on the Hilbert space tensor product, which is defined as.

• For the Hilbert space tensor product of the usual rates apply on commutativity, associativity and distributivity, that is, you have the following isometric isomorphisms, where the Hilbert spaces with elements are:

• The Hilbert space tensor has the so-called cross - norm property, which means that it applies

Construction as linear operators

For the tensor product and can be understood in terms of the dyadic product as a linear operator. The ( algebraic ) linear span of these operators is the algebra of operators of finite rank, it follows from the set of Fréchet - Riesz, based on this identification with the tensor product. The above defined scalar product induced just the Hilbert-Schmidt norm and the operators of finite rank are respect to this norm dense in the Hilbert - Schmidt operators, which are complete with respect to this norm. That is, the above carried out to complete the operators of finite rank as nothing else than the space of Hilbert-Schmidt opera gates from to.

Examples

• Be and the L2 - spaces to finite measure spaces. Then the Hilbert space tensor product is isomorphic to the space of the product of measure spaces, ie

• Let and be two sets and and the associated Hilbert spaces with orthonormal bases respectively. Then, the Hilbert space tensor is isomorphic to, that is, in the formulas

Tensor as orthogonal sums

Let and be Hilbert spaces and let be an orthonormal basis of. Then

An isometric isomorphic to subspace, and it is

With the right side is to be read as an orthogonal sum. The roles of, and you can swap of course. In this sense, a Hilbert space tensor product is nothing more than a suitable direct sum of copies of the two factors of the tensor product.

Operators on tensor products

Continuous linear operators and Hilbert spaces and can be assembled to the tensor on. More precisely:

The algebraic tensor product is continuous with respect to the pre-Hilbert space norm and can therefore be continued to a continuous linear operator. Where, with the operator norm on the left is.

This is the main motivation for the introduction of tensor products of Hilbert spaces. By means of these operators, one can define a tensor product of von Neumann algebras.

Comparison of different tensor

We consider tensor products of with itself Each element of the algebraic tensor product gives rise to a finite-dimensional operator, that is, the algebraic tensor product is included in a natural way. Denote the injective and projective tensor product and, we obtain:

• C * - algebra of compact operators with the operator norm.
• H * of the Hilbert -Schmidt algebra operators with the Hilbert-Schmidt norm. In the below textbook by R.V. Kadison and JR Ringrose is the connection of the Hilbert space tensor product of the Hilbert - Schmidt operators in the foreground.
• Banach * - algebra of trace class operators with the track as the norm.

This is to find, among other things in the below textbook by R. shadows.