Hilbert space tensor product

The formation of Hilbert space tensor products , considered in the mathematical sub-area of functional analysis , is a method for assembling new Hilbert spaces from Hilbert spaces. A purely algebraic formation of the tensor product is not sufficient, since in general one does not get complete spaces in this way . The injective and projective tensor products investigated in Banach space theory do not lead to the desired result either, since Hilbert spaces are generally not obtained in this way, that is, the norms are not defined by a scalar product .

Although scalar products on –Hilbert spaces are not bilinear , but only sesquilinear , it should still be possible to extend them to algebraic tensor products of Hilbert spaces, because tensor products are made for bilinear mappings, so to speak. Then one would at least have a prehilbert dream that one only had to complete in order to obtain a Hilbert dream. Exactly this approach proves to be successful. In the following only complex Hilbert spaces are considered, which are more important for many applications. The construction of tensor products of real spaces is very similar and is even simpler in some details. ${\displaystyle \mathbb {C} }$

definition

Let it be and two - silver dreams. The scalar products are always denoted with , for clarification the name of the Hilbert space is added as an index if necessary. Then you can show: ${\displaystyle H}$ ${\displaystyle K}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \langle \cdot ,\cdot \rangle }$

On the algebraic tensor product there is exactly one sesquilinear form with the property ${\displaystyle H\odot K}$

${\displaystyle \langle x_{1}\otimes y_{1},x_{2}\otimes y_{2}\rangle _{H\otimes K}=\langle x_{1},x_{2}\rangle _{H}\cdot \langle y_{1},y_{2}\rangle _{K}}$  for everyone and . ${\displaystyle x_{1},x_{2}\in H}$ ${\displaystyle y_{1},y_{2}\in K}$

The completion of the Prähilbert space is called the Hilbert space tensor product from and and is denoted by. Some authors use for the algebraic tensor product and then write for the completion, others use for both and indicate possible ambiguities or use a different notation for the algebraic tensor product, as has been done in this article. ${\displaystyle (H\odot K,\langle \cdot ,\cdot \rangle _{H\otimes K})}$ ${\displaystyle H}$ ${\displaystyle K}$ ${\displaystyle H\otimes K}$ ${\displaystyle H\otimes K}$ ${\displaystyle H\,{\overline {\otimes }}\,K}$ ${\displaystyle H\otimes K}$

properties

• The Hilbert space tensor product can easily be extended by induction to the Hilbert space tensor product of a finite number of Hilbert spaces , where is defined as. ${\displaystyle H_{1},\ldots ,H_{n}}$ ${\displaystyle H_{1}\otimes \ldots \otimes H_{n}}$ ${\displaystyle (H_{1}\otimes \ldots \otimes H_{n-1})\otimes H_{n}}$

• The usual theorems about commutativity, associativity and distributivity apply to the Hilbert space tensor product, i.e. one has the following isometric isomorphisms , where the Hilbert spaces with elements are: ${\displaystyle H_{i}}$ ${\displaystyle x_{i}}$

${\displaystyle H_{1}\otimes H_{2}\cong H_{2}\otimes H_{1}}$ With ${\displaystyle x_{1}\otimes x_{2}\mapsto x_{2}\otimes x_{1}}$

${\displaystyle H_{1}\otimes (H_{2}\otimes H_{3})\cong (H_{1}\otimes H_{2})\otimes H_{3}}$ With ${\displaystyle x_{1}\otimes (x_{2}\otimes x_{3})\mapsto (x_{1}\otimes x_{2})\otimes x_{3}}$

${\displaystyle H_{1}\otimes (H_{2}\oplus H_{3})\cong (H_{1}\otimes H_{2})\oplus (H_{1}\otimes H_{3})}$ With ${\displaystyle x_{1}\otimes (x_{2}\oplus x_{3})\mapsto (x_{1}\otimes x_{2})\oplus (x_{1}\otimes x_{3})}$

• The Hilbert space tensor product has the so-called cross norm property, that is, it holds

${\displaystyle \|x\otimes y\|=\|x\|\cdot \|y\|}$for all vectors and from the Hilbert spaces. ${\displaystyle x}$ ${\displaystyle y}$

Construction as linear operators

For and the tensor product can be understood as a linear operator in the sense of the dyadic product . The (algebraic) linear envelope of these operators is the algebra of the operators of finite rank , this follows from the Fréchet-Riesz theorem , on which this identification with the tensor product is based. The scalar product defined above induces the Hilbert-Schmidt norm and the operators of finite order lie with respect to this norm close to the Hilbert-Schmidt operators which are complete with respect to this norm. That is, the completion of the operators of finite rank carried out above results in nothing other than the space of the Hilbert-Schmidt operas from to . ${\displaystyle x\in H}$ ${\displaystyle y\in K}$ ${\displaystyle x\otimes y\colon K\to H}$ ${\displaystyle K}$ ${\displaystyle H}$

Examples

• Let and let the L ² spaces become finite dimensional spaces . Then the Hilbert space tensor product is isomorphic to the -space of the product of the measure spaces , that is ${\displaystyle L^{2}(X_{1},\Sigma _{1},\mu _{1})}$ ${\displaystyle L^{2}(X_{2},\Sigma _{2},\mu _{2})}$ ${\displaystyle \sigma }$ ${\displaystyle L^{2}}$

${\displaystyle L^{2}(X_{1},\Sigma _{1},\mu _{1})\otimes L^{2}(X_{2},\Sigma _{2},\mu _{2})\cong L^{2}(X_{1}\times X_{2},\Sigma _{1}\otimes \Sigma _{2},\mu _{1}\times \mu _{2})}$

• Let and be two sets and and the associated Hilbert spaces with orthonormal bases or . Then the Hilbert space tensor product is isomorphic to , that is, in formulas ${\displaystyle I}$ ${\displaystyle J}$ ${\displaystyle \ell ^{2}(I)}$ ${\displaystyle \ell ^{2}(J)}$ ${\displaystyle (e_{i})_{i\in I}}$ ${\displaystyle (e_{j})_{j\in J}}$ ${\displaystyle \ell ^{2}(I\times J)}$

${\displaystyle \ell ^{2}(I)\otimes \ell ^{2}(J)\cong \ell ^{2}(I\times J)}$.

This is the case because the Hilbert-Schmidt operators are precisely the operators with square-summable matrix coefficients . Since, according to Fischer-Riesz's theorem, every Hilbert space is isomorphic to one with a suitable one , it follows for arbitrary Hilbert spaces and ${\displaystyle \ell ^{2}(I)}$ ${\displaystyle I}$ ${\displaystyle H}$ ${\displaystyle K:}$

${\displaystyle \mathrm {dim} (H\otimes K)\,=\,\mathrm {dim} (H)\cdot \mathrm {dim} (K)\,,}$

where for the Hilbert space dimension , d. H. the cardinality of each orthonormal basis of stands. ${\displaystyle \mathrm {dim} (H)}$ ${\displaystyle H}$

Tensor products as orthogonal sums

Let and let Hilbert spaces and be an orthonormal basis of . Then ${\displaystyle H}$ ${\displaystyle K}$ ${\displaystyle (y_{j})_{j\in J}}$ ${\displaystyle K}$

${\displaystyle H\otimes y_{j}:=\{x\otimes y_{j};x\in H\}\subset H\otimes K}$

too isometrically isomorphic a subspace, and it is ${\displaystyle H}$

${\displaystyle H\otimes K\cong \bigoplus _{j\in J}H\otimes y_{j}}$,

where the right side is to be read as an orthogonal sum . The roles of and can of course be swapped. In this sense, a Hilbert space tensor product is nothing more than a suitable direct sum of copies of one of the two factors of the tensor product. ${\displaystyle H}$ ${\displaystyle K}$

Operators on tensor products

Continuous linear operators and Hilbert spaces and can be combined to the tensor product on . More accurate: ${\displaystyle A\in L(H)}$ ${\displaystyle B\in L(K)}$ ${\displaystyle H}$ ${\displaystyle K}$ ${\displaystyle A\otimes B}$ ${\displaystyle H\otimes K}$

The algebraic tensor product is continuous with respect to the Prähilbert space norm and can therefore be continued to a continuous linear operator . The following applies , with the operator norm of on the left . ${\displaystyle A\odot B:H\odot K\rightarrow H\odot K}$ ${\displaystyle A\otimes B\,\in \,L(H\otimes K)}$ ${\displaystyle \|A\otimes B\|=\|A\|\cdot \|B\|}$ ${\displaystyle L(H\otimes K)}$

This is the most important motivation for introducing tensor products for Hilbert spaces. Using these operators , one can define a tensor product for Von Neumann algebras . ${\displaystyle A\otimes B}$

Comparison of different tensor products

We consider tensor products of with itself. Every element from the algebraic tensor product gives rise to a finite-dimensional operator , that is, the algebraic tensor product is naturally contained in. Denoting and the injective or projective tensor product, we get: ${\displaystyle \ell ^{2}}$ ${\displaystyle \textstyle t=\sum _{i=1}^{n}x_{i}\otimes y_{i}}$ ${\displaystyle \textstyle T_{t}\colon \ell ^{2}\rightarrow \ell ^{2},\,x\mapsto \sum _{i=1}^{n}\langle x,y_{i}\rangle x_{i}}$ ${\displaystyle L(\ell ^{2})}$ ${\displaystyle \otimes _{\varepsilon }}$ ${\displaystyle \otimes _{\pi }}$

• ${\displaystyle \ell ^{2}\otimes _{\varepsilon }\ell ^{2}\,\cong }$ C * algebra of compact operators with the operator norm .
• ${\displaystyle \ell ^{2}\otimes \ell ^{2}\cong }$ H * -algebra of the Hilbert-Schmidt operators with the Hilbert-Schmidt norm. In the textbook by RV Kadison and JR Ringrose given below , the connection of the Hilbert space tensor product to the Hilbert-Schmidt operators is in the foreground.
• ${\displaystyle \ell ^{2}\otimes _{\pi }\ell ^{2}\cong }$ Banach - * - algebra of trace class operators with trace as norm.

This can be found, among other things, in the textbook by R. Schatten given below .

Individual evidence

1. ^ RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras , 1983, ISBN 0123933013 , Example 2.6.11
2. ^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras , 1983, ISBN 0123933013 , example 2.6.10
3. ^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras , 1983, ISBN 0123933013 , note 2.6.8
4. ^ Jacques Dixmier : Von Neumann algebras. North-Holland, Amsterdam 1981, ISBN 0-444-86308-7 , I.2.3

literature

• Richard V. Kadison , John R. Ringrose : Fundamentals of the Theory of Operator Algebras. Volume 1: Elementary Theory. Academic Press, New York NY 1983, ISBN 0-12-393301-3 ( Pure and Applied Mathematics 100, 1).
• Robert Schatten : A theory of cross-spaces. Princeton University Press, Princeton NJ 1950 ( Annals of Mathematical Studies 26, ISSN  0066-2313 ).