How to compare the absolute values of operator sums and the sums of absolute values? (Q2914878)

Given a matrix \(M_n(\mathbb C)\), let \(|A|=(AA^*)^{1/2}\). It is know that, for any \(A,B\in M_n(\mathbb C)\), there exist unitary matrices such that \(|A+B|\leq U |A|U^* + V|B|V^*\). The purpose of the present note is to compare \(|A+B|\) and \(|A|+|B|\). The following result is established. Let \(A_1, \dotsc, A_m\) with condition numbers dominated by~\(\omega>0\). Then NEWLINE\[NEWLINE |A_1+\dots +A_m|\leq \frac{\omega+1}{2\sqrt\omega} \bigl(|A_1|+\dots+|A_m|\bigr). NEWLINE\]NEWLINE The folowing conjecture is made: \(\bigl\| |A+B|\bigr\|_2\leq \sqrt {\frac {1+\sqrt 2}2}\bigl\||A|+|B|\bigr\|_2\), where \(\|\cdot\|_2\) is the Frobenius norm.