# Representing linear operators on infinite sequences as infinite matrices

So this question arose while I was working on a homework assignment to find the adjoint operator of a continuous operator on $l^2$. Anyway, it seemed like I could find it if I thought about the respective operators as infinite matrices. However, this is something that hasn't been discussed in class and I haven't found anything about on the internet. Is this valid? I've never really thought about infinite matrices before so I am just a little hesitant about what may be different from the finite cases I am use to. Can a linear operator on a sequence always be viewed as an infinite sequence?

Can anybody point me in the direction of a website or suggest a book that would give a good introduction to this? Thank you!

• math.stackexchange.com/questions/4483/… Commented Apr 28, 2013 at 14:10
• @Samuel, thanks, I guess I didn't search very well Commented Apr 28, 2013 at 14:18

Yes, any bounded linear operator in separable Hilbert space (and, in particular, in $\ell^2$) can be viewed as an infinite matrix. Also, some rules for finite matrices work in the case of infinite matrices. For instance, in order to find the matrix of the sum of bounded operators you can just find the sum of the matrices of these operators. Another example: the matrix of the adjoint operator of a continuous operator in Hilbert space is the adjoint matrix (conjugate traspose) of the matrix of this operator.