Bohr's inequalities for Hilbert space operators

The classical Bohr's inequality states that|z+w|2⩽p|z|2+q|w|2|z+w|2⩽p|z|2+q|w|2 for all z,w∈Cz,w∈C and all p,q>1p,q>1 with 1p+1q=1. In this paper, Bohr's inequality is generalized to the context of Hilbert space operators for all positive conjugate exponents p,q∈Rp,q∈R. In particular, the parallelogram law is recovered and some other interesting operator inequalities and established.