Re: a question for math guys

From: Stephen Montgomery-Smith (stephen_at_math.missouri.edu)
Date: 01/15/05 

Date: Sat, 15 Jan 2005 00:02:57 GMT

Gary Z. wrote:
> Hi, guys,
>
> I need some help on a math question.
>
> Here is my question:
>
> suppose that I have some infinite subspace in l^2 (square summable) space and, at the first hand, I can formulate my problem approximately in a finite subspace in l^2. Say, in an N-dimensional subspace, I know how to construct a matrix operation like:
>
> A_N = [ a_{i,j} ]
> 1 <= i, j <= N
>
> My ultimate intention would be to let N --> Infinity to span that infinite subspace of l^2. Doing so I would like to find the orthonormal basis that diagonalizes "A" meant by
>
> A := lim_{N-->Infinity} A_N
>
> however a matrix written as an infinite block has no practical use.
>
> Which part of the math theory, say, operator theory or functional analysis, "specifically" can help me have a concrete approach/procedure to accomplish my formulation that is solvable? What part of the operator theory, or else, should I seek for? Any hints would be appreciated.
>
> Thanks much in advance.
>
>

This kind of stuff can definitely be done, but there are just so many
different ways to define the limit, depending upon what your A_N are
(and also how A_N relates to say A_{N+1}). You want to start with
learning about the spectral decomposition of hermitian operators on
Hilbert spaces - I like the book by Kreizig (spelling?) on Functional
Analysis, which gives a very readable introduction. Depending on your
particular situation, you might end up studying von-Neumann algebras.
But believe me, this is a very difficult subject - no one really knows
about them in full generality.

My guess is that you should specify more precisely what you wish to do.
  Then we can provide advice more suited to your particular situation,
which may turn out to be a great deal easier than studying von Neumann
algebras.

Stephen