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Suppose we have a bounded, self-adjoint operator $T$ on a set of functions $\mathcal{F}$. What kinds of methods are there to find the spectrum of $T$?

Here is the setting I'm wondering about: consider a Markov process $X_t$, and suppose $P_t$ is the Markov semigroup which acts on measures $\mu(x)$ in the state space. The spectrum of $P_t$ relates to the rate of convergence of $\mu(x)$ to the invariant measure, and we're usually interested in the spectral gap. However I don't know how one can find the spectrum (or bounds on it, or the spectral gap) in general.

If $X_t$ is a discrete Markov chain, and we know all the transition probabilities, then the eigenvalues can be computed for the matrix $T$. But I don't know how to study the continuous version, where instead of a transition matrix we have an operator on functions.

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    $\begingroup$ It's usually hard to compute the spectrum explicitly except in very special cases. However there is a huge literature on ways to compute bounds. Heat Kernels and Spectral Theory by E.B. Davies might be one place to start. $\endgroup$
    – Nate Eldredge
    Commented May 14, 2021 at 15:09
  • $\begingroup$ If you have an explicit form of the generator of the semigroup (e.g., the Ornstein-Uhlenbeck operator), then I think semigroup methods work (maybe hard sometimes), e.g., spectral mapping theorem. $\endgroup$
    – S. Maths
    Commented May 21, 2021 at 19:10

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