Bohr's inequalities for Hilbert space operators (Q852749)

Let \(\mathbb{H}\) be a complex separable Hilbert space and \(B(\mathbb{H})\) the algebra of all bounded linear operators on \(\mathbb{H}\). For any \(X\in B( \mathbb{H}),\) write \(\left| X\right| =\left( X\ast X\right) ^{1/2}.\) The authors prove the following generalization of Bohr's inequality: for any \(A,B\in B(\mathbb{H})\) and any \(p,q\in \mathbb{R}\) with \(p>1\) and \(\frac{1}{p }+\frac{1}{q}=1,\) \(\left| A+B\right| ^{2}\leq p\left| A\right| ^{2}+q\left| B\right| ^{2}. \)