Operator valued analogues of multidimensional Bohr's inequality

TL;DR

This paper extends Bohr's inequality to operator-valued functions in multiple dimensions, providing sharp bounds and analogues for bounded analytic functions on complex domains.

Contribution

It establishes new multidimensional operator-valued Bohr inequalities, refining previous results and extending them to broader classes of functions and domains.

Findings

Sharp improved versions of Bohr's inequality for operator-valued functions.

Multidimensional analogues of operator-valued Bohr's inequality for complex domains.

Extensions of inequalities to functions in $H^{ abla}( ext{domain}, ext{operators})$.

Abstract

Let $\mathcal{B}(\mathcal{H})$ be the algebra of all bounded linear operators on a complex Hilbert space $\mathcal{H}$. In this paper, we first establish several sharp improved and refined versions of the Bohr's inequality for the functions in the class $H^{\infty}(\mathbb{D},\mathcal{B}(\mathcal{H}))$ of bounded analytic functions from the unit disk $\mathbb{D}:=\{z \in \mathbb{C}:|z|<1\}$ into $\mathcal{B}(\mathcal{H})$. For the complete circular domain $Q \subset \mathbb{C}^n$, we prove the multidimensional analogues of the operator valued Bohr's inequality established by G. Popescu [Adv. Math. 347 (2019), 1002-1053]. Finally, we establish the multidimensional analogues of several improved Bohr's inequalities for operator valued functions in $Q$.