I am working with a class of matrices $A$ which are non-negative-definite, not symmetric, and have maximum eigenvalue less than 1. I am interested in the spectral properties of the matrix $H = A(I - A)^{-1}$.

Is there a spectral theorem for this context, which gives sufficient conditions on $A$ for the matrix $H$ to be diagonalizable? Are there sufficient conditions on $A$ to guarantee that the eigenvalues of $H$ decay rapidly (e.g., exponentially)?

I am sure that such questions have been analyzed in the past, perhaps in the economics literature. However, I have been unable to find a reference. While I phrased these question in terms of matrices, I of course would also be interested in the more general context of linear operators on Hilbert spaces.