Further unitarily invariant norm inequalities for positive semidefinite matrices

Abstract

In this paper, we prove further unitarily invariant norm inequalities for positive semidefinite matrices. These inequalities generalize earlier related inequalities. Among other applications of our new inequalities, we prove that

$$\begin{aligned} \left\| XZ+ZY\right\| \le \max \left( \left\| X\right\| ,\left\| Y\right\| \right) \left\| Z\right\| +\frac{1}{2} \left\| X^{*}Z+ZY^{*}\right\| \end{aligned}$$

for all n $\times$ n complex matrices X, Y,  and Z. Here, $\left\| \cdot \right\|$ denotes the spectral norm.