Generalizations of Bohr inequality for Hilbert space operators (Q1025033)

Let \({\mathcal B}({\mathbb H})\) be the space of all bounded linear operators on a complex separable Hilbert space \({\mathbb H}.\) Bohr's inequality for Hilbert space operators asserts that, for \(A, B\in {\mathcal B}({\mathbb H})\) and \(p, q>1\) real numbers such that \(1/p+1/q=1,\) \[ |A+B|^2\leq p|A|^2+q|B|^2 \] with equality if and only if \(pA=qB.\) In this paper, a number of generalizations of Bohr's inequality for operators in \({\mathcal B}({\mathbb H})\) are established. Moreover, Bohr inequalities are extended to multiple operators and some related inequalities are obtained. For each inequality a necessary and sufficient condition for the equality case is determined. The results obtained in this paper generalize previous existing results in the literature.