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When is the resolvent of a sum of operators the sum of the resolvents of these operators?

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I'm talking of the top of my head and am by no means an expert but the only way that you could do anything like that, i guess, is if you knew something about the spectrum of both of them. I mean the worst guess that can have is to look at the norm of both of them, take the bigger one, and you can be sure your resolvent will be defined outside the sphere with that radius.

If one of them is unbounded i have no clue but would be interested in the answer of someone who has.

If what i said is false please enlighten me.
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I could give a better answer if I actually knew what a resolvent was, but from cursory googling it has something to do with the spectrum of an operator. I’m gonna pretend this question is about eigenvalues of a linear operator on a finite dimensional vector space, and hope that’s close enough that you can translate back to the actual situation.

Eigenvalues don’t behave well with respect to addition of operators, so I wouldn’t expect knowing about each separately would tell you anything about the sum. On the other hand, if your operators commute, you can simultaneously diagonalize them, and then their eigenvalues would pair up and add. Unfortunately I don’t think your operators commute unless dq/dx = 0, so that’s not super helpful.

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