Operator valued analogues of multidimensional refined Bohr's inequalities

Abstract: Let $\mathcal{B}(\mathcal{H})$ denote the Banach algebra of all bounded linear operators acting on complex Hilbert spaces $\mathcal{H}$. In this paper, we first establish several sharply refined versions of Bohr's inequality analogues with operator valued functions in the class $\mathcal{B}(\mathbb{D}, \mathcal{B}(\mathcal{H}))$ of bounded analytic functions from the unit disk $\mathbb{D}$ to $\mathcal{B}(\mathcal{H})$ with $\sup_{|z|<1}\Vert f(z)\leq 1$ by utilizing a certain power of the function's norm. Additionally, we establish several multidimensional analogues of refined Bohr's inequalities by using operator valued functions in the complete circular domain $\Omega\subset\mathbb{C}^n$. All of the results are sharp.