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I stumbled over the eigenvalue problem while analysing an infinite-dimensional jump process. The state space I am working with is $$ \mathbb{N}^{<\infty}=\bigcup_{k=1}^{\infty}\mathbb{N}^k $$ i.e. all finite positive integer tuples. The Q-matrix of the process has the following form: after some exponential time we either increase one of the entries in the current tuple by one or append 1 to the end of the current tuple, so we are in a birth-only setting.

As one can see this process is neither irreducible nor recurrent and also we are not in a finite state space. My question is, if there is anything known about the eigenvalues/-functions for such a Q-matrix?
Basically I am interested in solutions of the equation $$ Qf = \lambda f \quad $$ given some $\lambda$.

Thanks a lot!

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