I was reading the book Matrix Analysis and have encountered with the following exercise in Chapter IX. A Selection of Matrix Inequalities:
Let $X, Y$ be positive operators. Show that for every unitarily invariant norm $$ \| X - Y \| \le \left\| \begin{pmatrix} X & 0 \\ 0 & Y \end{pmatrix} \right\|. $$
After some time that I spend proving it I started to plug-in different values for $X$ and $Y$ trying to understand the behaviour of the norms of these matrices. So, I picked the operator norm, which is indeed unitarily invariant norm.
Now, take positive $2\times 2$ matrices $$ X = \begin{pmatrix} 1 & 1 \\ 0 & 0.5 \end{pmatrix}, \quad Y = \begin{pmatrix} 0.5 & 0 \\ 1 & 1\end{pmatrix}. $$ I calculated and triple double checked that the inequality above is violated with r.h.s. $\approx 1.46$ and l.h.s. $1.5$.
My question is: Can anyone verify whether this "counterexample" is true or not? If not, where is the mistake?
UPD: As it was correctly pointed out by @Ted Shifrin the operators should be Hermitian (symmetric) and the counterexample fails. @amsmath gives the proof for operator norm in comments. Any ideas how it can be proved for any unitarily invariant norm?