Let $k\in\mathbb{N}$ and let $d\in\mathcal{M}_k=\mathcal{B}(\ell_2^k)$ be a $k\times k$ diagonal matrix with strictly positive entries. Let now $x_1,\ldots,x_m\in\mathcal{M}_k\,\,(m\in\mathbb{N}$ - arbitrary) be a finite family of self-adjoint $k\times k$ matrices. Is it true that $\|\sum_{j=1}^m(dx_jd)^2\|_{\mathcal{M}_k}\leqslant\|\sum_{j=1}^md^2x_j^2d^2\|_{\mathcal{M}_k}$? I would be happy to have this inequality up to a constant independent of $k$ and $m$. This question might be connected with the following one (we keep the same notation): Let $x=\begin{pmatrix} x_1 & \ldots & x_m \\ 0 & \ldots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \ldots & 0 \end{pmatrix}$ be a block matrix the first row of which consists of the $m$ previously chosen self-adjoint $k\times k$ matrices and each of the remaining $m-1$ rows consists of $k\times k$ zero matrices. In particular $x\in\mathcal{M}_{mk}$. We now create a bolck-circulant matrix
$y=\begin{pmatrix} x_1 & x_2 & \ldots & x_m \\ x_2 & x_3 & \ldots & x_1 \\ x_3 & x_4 & \ldots & x_2 \\ \vdots & \vdots & \ddots & \vdots \\ x_m & x_1 & \ldots & x_{m-1} \end{pmatrix}$. Of course $y$ is a self-adjoint $mk\times mk$ matrix. Can we express the norm $\|y\|_{\mathcal{M}_{mk}}$ in terms of the norms $\|x_j\|_{\mathcal{M}_k}$?
