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Feb 15, 2011 at 8:31 comment added András Bátkai Another good sourse to your problem might be the book Weidmann: Linear operators in Hilbert spaces, Springer. I belive, in research we all have to learn all the time.
Feb 15, 2011 at 1:42 comment added psyduck I'll check out the "short course on operator semigroups" as you suggested. In the mean-time, I think I have found an answer to my quandary (on wikipedia embarrassingly). I can just write my solution as $\int e^{-\lambda t} \phi_\lambda \mu( d\lambda )$ where $\mu$ is the spectral measure. At least, I think this works (I'm a physicist ... we aren't normally taught things like this in our coursework--at least I never was--so we need to learn this as we go in research). In any case, thank you very much for your reply to my question.
Feb 15, 2011 at 1:36 vote accept psyduck
Feb 14, 2011 at 23:08 history answered András Bátkai CC BY-SA 2.5