The generalized characteristic polynomial, corresponding resolvent and their application

We introduced previously the generalized characteristic polynomial defined by $P_C(\lambda )=\mathrm{d}\mathrm{e}\mathrm{t}C(\lambda ),$ where $C(\lambda )=C+\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}({\lambda }_1,\dots ,{\lambda }_n)$ for $C\in \mathrm{M}\mathrm{a}\mathrm{t}(n,\mathbb{C})$ and $\lambda =({\lambda }_k{)}_{k=1}^n\in {\mathbb{C}}^n$ and gave the explicit formula for $P_C(\lambda )$. In this article we define an analogue of the resolvent $C(\lambda {)}^{-1}$, calculate it and the expression $(C(\lambda {)}^{-1}a,a)$ for $a\in {\mathbb{C}}^n$ explicitly. The obtained formulas and their variants were applied to the proof of the irreducibility of unitary representations of some infinite-dimensional groups.