Bohr's Inequality for Non-commutative Hardy Spaces

TL;DR

This paper generalizes Bohr's inequality to non-commutative Hardy spaces associated with von Neumann algebras, providing new bounds for operators and matrices of various sizes.

Contribution

It extends classical Bohr's inequality to non-commutative settings and determines optimal bounds for different matrix sizes and operator classes.

Findings

Optimal bound for von Neumann-Shcatten class is 1/3.

Bound for 2x2 matrices is 1/2.

Bound for 3x3 matrices is approximately 0.414.

Abstract

In this paper, we extend the classical Bohr's inequality to the setting of the non-commutative Hardy space $H^1$ associated with a semifinite von Neumann algebra. As a consequence, we obtain Bohr's inequality for operators in the von Neumann-Schatten class $\cl C_1$ and square matrices of any finite order. Interestingly, we establish that the optimal bound for $r$ in the above mentioned Bohr's inequality concerning von Neumann-Shcatten class is 1/3 whereas it is 1/2 in the case of $2\times 2$ matrices and reduces to $\sqrt{2}-1$ for the case of $3\times 3$ matrices. We also obtain a generalization of our above-mentioned Bohr's inequality for finite matrices where we show that the optimal bound for $r$, unlike above, remains 1/3 for every fixed order $n\times n,\ n\ge 2$.