  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 $Q$.
