Kronecker product of infinite dimensional matrices

Hi!

Suppose we are given an infinite dimensional vector space V with a basis satisfying the following property: all the elements in V can be represented by a finite linear combinations of the elements of the basis. Now I should be able to talk about linear maps in this space, and their matrix representation, e.g.

A:V→V,B:V→V,

where AB or BA are well defined too. This is what I understand from reading Lectures in Abstract Algebra, Vol. 2 by Nathan Jacobson (http://reslib.com/book/Lectures_in_A...ebra__Vol__2#1), chapter 9 on infinite dimensional vector spaces.

From here, I want to talk about the matrix that represents the kronecker product A⊗B, but I'm not sure if it is well defined.

The interesting thing is, that later in chapter 9 of this book (page 256 middle paragraph) the author says that we can talk about Kronecker product of arbitrary vector spaces. However, at that point I could not follow the details due to insufficient background, so I don't know if it indeed answers my question.

Finally, if such an infinite matrix is well defined, I'd like to know if it satisfies the Kronecker product properties that could be found in wikipedia on Kronecker product: bilinearity, associativity and the mixed-product property.

Thank you for your help.

Re: Kronecker product of infinite dimensional matrices

Hey yphink.

The question you want to ask your yourself is whether you get all the required well-defined operator properties like convergence in l^2 in all the expected ways (since you have infinity-rank you are going to get lots of situations where you get convergence in one sense but not in the another).

You will start by proving whether the sum of two vectors in l^2 is in the same space (which it should be) and then apply that to what is happening within the product (since you get a11xb11 in the operator which means you need to consider whether a1*B1 is well defined and when the summation of all these when operating on some vector is also well defined).

I'd start off with this first and then look at the other properties later.