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).