Operator valued analogues of multidimensional Bohr's inequality

Let B(H) be the algebra of all bounded linear operators on a complex Hilbert space 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-infinity(D, B(H)) of bounded analytic functions from the unit disk D := {z is an element of C : vertical bar z vertical bar < 1} into B(H). For the complete circular domain Q subset of C-n, we prove the multidimensional analogues of the operator valued Bohr-type inequality which can be viewed as a special case of the result by G. Popescu [Adv. Math. 347 (2019), 1002-1053] for free holomorphic functions on polyballs. Finally, we establish themultidimensional analogues of several improved Bohr's inequalities for operator valued functions in Q.