Operator valued analogues of multidimensional refined Bohr's inequalities

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 ${\sup }_{\mathrm{\mid }z\mathrm{\mid }<1}\mathrm{\parallel }f(z)\le 1$ 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 $\mathrm{\Omega }\subset {\mathbb{C}}^n$. All of the results are sharp.