Spectrum of an infinite matrix?

Hello all,

I am a graduate student, although unfortunately not in mathematics, and I have a peculiar math problem which has stumped me! Basically the situation is I have a first-order Markov process, so state vector v_n+1 = M * v_n, where M is a left-stochastic transition matrix that carries you from vector to vector. However I would like to modify this model so that there is no cap on the vectors (i.e. they have infinite elements but still sum to 1). The transition matrix takes an unusual form where you have some distribution (binomial, although I don't think that's important for what I'm asking) that describes the probability of moving from state vector element m to element n (column and row of the matrix, respectively), and for all possible initial state vector elements, there are 2x that many possible final state vector elements. So for a uniform distribution, the matrix looks something like (sorry if this doesn't line up correctly):

1 1/2 1/4 1/4 1/6 ...

0 0 0 0 0

0 1/2 1/2 1/4 1/6

0 0 0 0 0

0 0 1/4 1/4 1/3

0 0 0 0 0

0 0 0 1/4 1/6

0 0 0 0 0

0 0 0 0 1/6

...

So specifically the problem I'm having is that I want to find the spectrum for this infinite matrix (actually, just the highest 2 eigenvalues). I was hoping to just diagonalize for finite-dimensional version and extrapolate. The trouble is that for any (finite) maximum state vector value this is not a square matrix. So I'm at a loss! If anyone could point me in the right direction, I'd really appreciate it.

Thanks!

- George