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

Abstract

We show that if $X$ and $Y$ are two non-zero Hilbert--Schmidt operators, then for any $\lambda\geq0$,
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\begin{align*}
&\cos^2\Theta_{_{X,Y}}
\\& \,\, \leq \frac{1}{1+\lambda}\sqrt{\cos\Theta_{_{|X^*|,|Y^*|}}}\sqrt{\cos\Theta_{_{|X|,|Y|}}}\,\frac{|\langle X, Y\rangle|}{{\|X\|}_{_2}{\|Y\|}_{_2}}
+ \frac{\lambda}{1+\lambda}\cos\Theta_{_{|X^*|,|Y^*|}}\cos\Theta_{_{|X|,|Y|}}
\\& \,\, \leq \cos\Theta_{_{|X^*|,|Y^*|}}\cos\Theta_{_{|X|,|Y|}}.
\end{align*}
\endgroup
Here $\Theta_{_{A,B}}$ denotes the angle between non-zero Hilbert--Schmidt operators
$A$ and $B$. This enables us to present some inequalities for the Hilbert--Schmidt norm.
In particular, we prove that
\begin{align*}
{\big\|X + Y\big\|}_{_2} \leq \sqrt{\frac{\sqrt{2} + 1}{2}}\,{\big\|\,|X| + |Y|\,\big\|}_{_2}.
\end{align*}