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The object of the present work is to construct all the generalized spectral functions of a certain class of Carleman operators in the Hilbert space $L^2\left(X,\mu \right)$ and establish the corresponding expansion theorems, when the deficiency indices are (1,1). This is done by constructing the generalized resolvents of $A$ and then using the Stieltjes inversion formula.

In the present work, using a formula describing all scalar spectral functions of a Carleman operator $A$ of defect indices $(1,1)$ in the Hilbert space $L^2(X,\mu )$ that we obtained in a previous paper, we derive certain results concerning the localization of the spectrum of quasi-selfadjoint extensions of the operator $A$.