Lipschitz-type bounds for the map \(A \to|A|\) on \({\mathcal L}({\mathcal H})\) (Q1863534)

Let \(L(H)\) be the algebra of all bounded linear operators on an infinite-dimensional Hilbert space \(H\). For a positive real number \(R\), let \(B_R= \{A\in L(H):\|A\|> R\}\), where \(\|\cdot\|\) denotes the usual operator norm. Denote by \(B^+_R\) the subset of \(B_R\) consisting of all nonnegative operators. In this paper, the authors show the following: (1) If \(0< R< {1\over 16}\), then the square root map \(A\to A^{1/2}\) is Lipschitz continuous on the set \(B^+_R\), that is \(\|A^{1/2}- B^{1/2}\|\leq {\sqrt{8}\over R} \|A-B\|\) for all \(A\), \(B\in B^+_R\). (2) If \(0< R< {1\over 4}\), then the absolute value map \(A\to|A|\) is Lipschitz continuous on the set \(B_R\), that is \(\||A|-|B|\|\leq {4\sqrt{2}\over R} \|A- B\|\) for all \(A\), \(B\in B_R\).