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.