I am looking for some references related to the problem of compactness of the resolvent for a given unbounded operator. I do not assume that the operator is self-adjoint. (I found only few books and articles, but there this was assumed.) My problem is the following. I have a densely-defined operator $T$ on a separable Hilbert space and I know that there exist its adjoint (on the same domain) and I know also something about $F(TT^\ast)$, where the last in the spirit of functional calculus. I would like to say something about the resolvent of $T$ and I am looking for some reviews of that topic where there would be some criteria for compactness of the resolvent. (I have plenty of examples to analyse hence I do not specify this function $F$ here).

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    $\begingroup$ Differential operators are not nice, but integral resolvents are often compact on finite regions. The compactness is generally proved by showing that bounded subsets are mapped by the resolvent to equicontinuous families of functions. $\endgroup$ – DisintegratingByParts Jan 17 '18 at 18:03
  • $\begingroup$ Unfortunately they are differential (at least most of them). $\endgroup$ – mikis Jan 17 '18 at 18:27

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