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Sorry if the question sounds silly, but can a continuous matrix be seen as a differential operator?

First of all, let me state that I have no idea what a continuous matrix would formally mean, but I would suppose there is such an abstract notion, somewhere?

Secondly, let me tell you where I'm coming from: I'm reading about dynamic processes and it seems they can sometimes be described by the notion of a "generator". For dynamic processes with a finite state space, the generator is a (normal) matrix (e.g. discrete space Markov processes). For dynamic processes with a continuous state space (e.g. Hamiltonian dynamics), the generator is a differential operator. My above question follows naturally from this observation.

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# Continuous matrix = differential operator?

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