Singular value and unitarily invariant norm inequalities for matrices

In this paper, we prove some new singular value and unitarily invariant norm inequalities for matrices. Among other results, it is shown that if X, Y, Z, W are n \(\times \) $\times$ n matrices, then \(\begin{aligned} s_{j}\left( XY+ZW\right) \le \textrm{max}\left( \left\| Y\right\| ,\left\| Z\right\| \right) s_{j}\left( X\oplus W\right) +\frac{1}{2} \left\| XY+ZW\right\| \end{aligned}\) $$\begin{array}{c}\hfill s_j\left(X,Y,+,Z,W\right)\le \text{max}\left(∥Y∥,,,∥Z∥\right)s_j\left(X,\oplus ,W\right)+\frac{1}{2}∥X,Y,+,Z,W∥\end{array}$$ and \(\begin{aligned} \Vert XY\pm YX\Vert \le \Vert X\Vert \Vert Y\Vert +w(XY) \end{aligned}\) $$\begin{array}{c}\hfill \Vert XY\pm YX\Vert \le \Vert X\Vert \Vert Y\Vert +w(XY)\end{array}$$ for \(j=1,2,\ldots ,n\) $j=1,2,\dots ,n$ , where \(\left\| \cdot \right\| ,w(\cdot ),\) $∥·∥,w(·),$ and \( s_{j}(\cdot )\) $s_j(·)$ denote the spectral norm, the numerical radius, and the jth singular value of matrices.