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arXiv · 2310.17351

The generalized characteristic polynomial, corresponding resolvent and their application

A. V. Kosyak
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Abstract

We introduced previously the generalized characteristic polynomial defined by $P_C(λ)={\rm det}\,C(λ),$ where $C(λ)=C+{\rm diag}\big(λ_1,\dots,λ_n\big)$ for $C\in {\rm Mat}(n,\mathbb C)$ and $λ=(λ_k)_{k=1}^n\in \mathbb C^n$ and gave the explicit formula for $P_C(λ)$. In this article we define an analogue of the resolvent $C(λ)^{-1}$, calculate it and the expression $(C(λ)^{-1}a,a)$ 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.

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A. V. Kosyak. 2023-10-26. The generalized characteristic polynomial, corresponding resolvent and their application. https://arxiv.org/abs/2310.17351

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