On a norm inequality for a positive block-matrix

TL;DR

This paper investigates a specific norm inequality for positive semidefinite block matrices, establishing conditions under which the inequality ||H|| ≤ ||A+B|| holds and exploring related topics.

Contribution

The paper identifies conditions ensuring the norm inequality for positive block matrices and explores related matrix inequalities.

Findings

The inequality ||H|| ≤ ||A+B|| holds under certain conditions.

Conditions for the inequality depend on properties of the blocks A, B, and X.

Related topics include extensions and limitations of the inequality.

Abstract

For a positive semidefinite matrix $H= \begin{bmatrix} A&X\\ X^{*}&B \end{bmatrix} $, we consider the norm inequality $ ||H||\leq ||A+B|| $. We show that this inequality holds under certain conditions. Some related topics are also investigated.