Direct product of operators

In summary, the properties of tensor product for operators in infinite-dimensional Hilbert spaces are similar to those in finite-dimensional Hilbert spaces, with some additional care needed. These properties can be found in various books on Operator Theory, such as "Lectures on von Neumann Algebras" by Serban Stratila and Laszlo Zsido, and "Operator algebras and quantum statistical mechanics" by Bratteli and Robinson.
  • #1
LagrangeEuler
717
20
Is this correct in infinite dimensional Hilbert spaces?
## (\hat{A}_1 \otimes \hat{A}_2)^{-1}=\hat{A}^{-1}_1 \otimes \hat{A}^{-1}_2 ##
## (\hat{A}_1 \otimes \hat{A}_2)^{\dagger}=\hat{A}^{\dagger}_1 \otimes \hat{A}^{\dagger}_2 ##
## (\hat{A}_1 +\hat{A}_2) \otimes \hat{A}_3=(\hat{A}_1 \otimes \hat{A}_2)+(\hat{A}_2 \otimes \hat{A}_3) ##
## \hat{A}_1 \otimes (\hat{A}_2+\hat{A}_3)=(\hat{A}_1 \otimes \hat{A}_2)+(\hat{A}_1 \otimes \hat{A}_3) ##
## \hat{1} \otimes \hat{1}=\hat{1} ##
## (\hat{A}_1 \otimes 0)=(0 \otimes \hat{A}_2)=0 ##
Can you tell me a book where I can see this properties. I found this only for operators which acts in finite dimensional Hilbert spaces.
 
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  • #2
Your question belongs to the part of mathematics usually discussed within "Operators theory". But here is essentially the positive answer to your questions (with somewhat different notation) - provided appropriate care is being taken:

stratila58.jpg


Taken from "Lectures on von Neumann Algebras" by Serban Stratila and Laszlo Zsido, Abacus Press 1975. You can also find it in online Bratteli and Robinson book "Operator algebras and quantum statistical mechanics", Vol. 1, but there it is more complicated as tensor product of not just two but a family of Hilbert spaces is being considered.
 

1. What is the direct product of operators?

The direct product of operators is a mathematical operation that combines two operators to create a new operator. It is denoted by the symbol ⊗ and is used in linear algebra and quantum mechanics.

2. How is the direct product of operators calculated?

The direct product of operators is calculated by taking the tensor product of the matrices that represent the individual operators. This is done by multiplying each element of one matrix with the entire other matrix.

3. What is the significance of the direct product of operators?

The direct product of operators is significant because it allows us to describe the behavior of two operators acting on a system simultaneously. It is also used to represent the composite system of two subsystems.

4. Can the direct product of operators commute?

In general, the direct product of operators does not commute, meaning that the order in which they are multiplied matters. However, in some cases, such as when the operators are diagonal, they can commute.

5. How is the direct product of operators related to the Kronecker product?

The direct product of operators and the Kronecker product are closely related, as the Kronecker product is a generalization of the direct product for matrices. The Kronecker product is used when the operators act on different vector spaces, while the direct product is used when they act on the same vector space.

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