Title : ( A quadratic operator equation and a non-commutative uncertainty relation )
Authors: Fateme Abdollahzade , Hamed Najafi ,
دانلود فایل برای اعضای دانشگاهAccess to full-text not allowed by authors
Abstract
In this paper, we explore equivalent conditions for the positivity of a quadratic equation $\\\\\\\\textbf{P}(x) = Ax^2 + Bx + C$ with operator coefficients. Specifically, we show that $Ax^2 + 2Bx + C \\\\\\\\geq 0$ for all $x \\\\\\\\in \\\\\\\\mathbb{R}$ if and only if $A, C \\\\\\\\geq 0$, $B = B^*$, and there exists a self-adjoint operator $H$ such that $$\\\\\\\\begin{bmatrix} A+C+H & B+i(A-C) \\\\\\\\\\\\\\\\ B-i(A-C) & A+C-H \\\\\\\\end{bmatrix}\\\\\\\\geq 0.$$
Keywords
quadratic equation; geometric mean; numerical range
@article{paperid:1103761,
author = {Abdollahzade, Fateme and Najafi, Hamed},
title = {A quadratic operator equation and a non-commutative uncertainty relation},
journal = {Linear and Multilinear Algebra},
year = {2025},
volume = {73},
number = {8},
month = {May},
issn = {0308-1087},
pages = {1792--1803},
numpages = {11},
keywords = {quadratic equation; geometric mean; numerical range},
}
%0 Journal Article
%T A quadratic operator equation and a non-commutative uncertainty relation
%A Abdollahzade, Fateme
%A Najafi, Hamed
%J Linear and Multilinear Algebra
%@ 0308-1087
%D 2025
