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!


1 Answer 1


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.

However, in the infinite dimensional case, usually, it is easier and more convinient to deal with an operator itself than with its matrix. One of the main reasons is that, often, it's very hard to reformulate any topological properties of operator (such as continuity, compactness) in terms of corresponding matrix. Also, computations with infinite matrices are very cumbersome and they often overshadow main ideas of the theory. That's why it is difficult to find anything about infinite matrices in standart courses of functional analysis.

Although, matrices can be very helpful for the study of some specific classes of bounded linear operators. But it's much more advanced topics of functional analysis. You can find some information about these specific classes of operators in the book "Analysis of Toeplitz Operators" by A. Bottcher and B. Silberman.

When one consider unbounded linear operators everything gets much worse because even simple algebraical rules for the matrices of operators do not work in this case. Therefore matrices is totally useless in unbounded case.


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