A geometric approach to inequalities for the Hilbert--Schmidt norm

We define angle ${\mathrm{\Theta }}_{X,Y}$ between non-zero Hilbert–Schmidt operators $X$ and $Y$ by $\cos {\mathrm{\Theta }}_{{}_{X,Y}}=\frac{\mathrm{R}\mathrm{e}\mathrm{T}\mathrm{r}(Y^{\ast }X)}{{\mathrm{\parallel }X\mathrm{\parallel }}_{{}_2}{\mathrm{\parallel }Y\mathrm{\parallel }}_{{}_2}}$, and give some of its essentially properties. It is shown, among other things, that \begin{align*} \big|\cosΘ_{{X,Y}}\big|\leq \min\left{\sqrt{\cosΘ{{|X^|,|Y^|}}}, \sqrt{\cosΘ{{|X|,|Y|}}}\right}. \end{align*} It enables us to provide alternative proof of some well-known inequalities for the Hilbert–Schmidt norm. In particular, we apply this inequality to prove Lee’s conjecture [Linear Algebra Appl. 433 (2010), no.~3, 580–584] as follows \begin{align*} {\big|X + Y\big|}{2} \leq \sqrt{\frac{\sqrt{2} + 1}{2}},{\big|,|X| + |Y|,\big|}{_2}. \end{align*} A numerical example is presented to show the constant $\sqrt{\frac{\sqrt{2}+1}{2}}$ is smallest possible. Other related inequalities for the Hilbert–Schmidt norm are also considered.