Some spectral results for certain positive operators in Hilbert Spaces
Rashid A., P Sam Johnson
TL;DR
The paper addresses spectral properties of positive operators arising from $P$-matrices in Hilbert spaces and develops a factorization framework for $P$-matrices using eigenvalues. It leverages the transform $U(A)=(I+A)^{-1}(I-A)$ to relate $A$ to a product of $P$-matrices and proves a diagonal-scaling factorization $S^{-1}AT$ into positive-stable $P$-matrices, while also constructing a spectral theory for $P$-operators that yields positivity of real eigenvalues and a symmetric-function criterion for realizable spectra. Key contributions include a concrete factorization method for $P$-matrices, positivity results for spectra of $P$-operators, and a complete characterization of possible spectra via the positivity of all elementary symmetric functions $oldsymbol{\sigma_k}$. These results enhance operator-theoretic tools for stability analysis in linear complementarity problems and related areas of functional analysis and quantum mechanics.
Abstract
This paper investigates spectral properties of certain classes of positive operators originated from different matrices appeared in linear complementarity problem. These positive operators play a crucial role in various areas of mathematics and its applications, including operator theory, functional analysis, and quantum mechanics. Understanding their spectral behavior is essential for analyzing the dynamics and stability of systems governed by such operators. P-matrix is one of the important types of matrices appearing in linear complementarity problems. In this research, with the help of spectral results we have given a factorization for P-matrix, as the product of two non-trivial P-matrices. We also focus on elucidating spectral properties such as eigenvalues, approximate eigenvalues and spectral values associated with certain positive operators.