Diagonal and off-diagonal blocks of positive definite partitioned matrices

TL;DR

This paper investigates inequalities involving blocks of positive definite matrices and proposes a conjecture for a triangle inequality involving contractions, aiming to deepen understanding of matrix block structures and operator inequalities.

Contribution

It establishes a sharp inequality for blocks of positive definite matrices and introduces a conjecture on a triangle inequality for contractions, proposing a new direction in matrix analysis.

Findings

Established a sharp inequality for positive definite matrix blocks.

Conjectured a triangle inequality for contractions with a specific constant.

Proposed a new conjecture for operator inequalities involving contractions.

Abstract

We establish a sharp inequality between the blocks of positive partitioned matrices and conjecture a triangle type inequality for contractions: Given three contactions A,B,C, we conjecture that the constant c=3/4 is sharp in the triangle inequality |A+B+C| < cI + |A| +|B| + |C|.