norms of row matrices

Question

Let $x_1,\ldots,x_m,y_1,\ldots,y_m$ be $k\times k$ sqaure matrices and assume $\|x_j\|\leqslant\|y_j\|$ for all $j=1,\ldots,m$ (the norm in $B(\ell_2^k)$). Now define the block matrices $x,y\in M_{mk}$, the first rows of which are $x_j$'s and $y_j$'s, respectively and other rows consist of $k\times k$ zero matrices. Is it true that $\|x\|\leqslant\|y\|$ (the norm in $B(\ell_2^{mk})$)?

Answer 1

This is false. Let $$ x_1=x_2=\begin{bmatrix}1&0\\0&1\end{bmatrix},\ \ y_1=\begin{bmatrix}1&0\\0&0\end{bmatrix}\ \ y_2=\begin{bmatrix}0&0\\0&1\end{bmatrix}. $$ Then $\|x_1\|=\|x_2\|=\|y_1\|=\|y_2\|=1$, but $$ \left\|\begin{bmatrix}x_1& x_2\\0&0\end{bmatrix}\right\| = \left\|\begin{bmatrix}1&0&1&0\\0&1&0&1\\0&0&0&0\\0&0&0&0\end{bmatrix}\right\|=\sqrt2, $$ while $$ \left\|\begin{bmatrix}y_1& y_2\\0&0\end{bmatrix}\right\| = \left\|\begin{bmatrix}1&0&0&0\\0&0&0&1\\0&0&0&0\\0&0&0&0\end{bmatrix}\right\|=1. $$ The norms are very easy to calculate because $$ \left\|\begin{bmatrix}A&B\\0&0\end{bmatrix}\right\|^2 =\left\|\begin{bmatrix}A&B\\0&0\end{bmatrix}\begin{bmatrix}A^*&0\\B^*&0\end{bmatrix}\right\| =\left\|\begin{bmatrix}AA^*+BB^*&0\\0&0\end{bmatrix}\right\|=\|AA^*+BB^*\|. $$