Inequality on spectral norm of sum of tensor products

Question

I am looking for a lower bound on $$\left\|\sum\nolimits_i M_i \otimes M_i\right\|_{\infty}$$ when the $M_i$'s are positive semi-definite matrices and $\|.\|_{\infty}$ denote the spectral norm. I suspect that the inequality $\left\|\sum\nolimits_i M_i \otimes M_i\right\|_{\infty} \geq \left\|\sum\nolimits_i M_i^2\right\|_{\infty}$ that holds when the matrices commute is not true in general. I'm interested in any counter-example to that, and alternative inequalities.

Answer 1

Since you have $$ 0\leq M_k\otimes M_k\leq\sum_jM_j\otimes M_j, $$ and $\|M_k\|^2=\|M_k\otimes M_k\|$, you have the inequality $$ \|M_k\|^2\leq\Big\|\sum_jM_j\otimes M_j\Big\|, $$ which gives $$ \max_k\|M_k\|^2\leq\Big\|\sum_jM_j\otimes M_j\Big\|. $$ This bound is sharp. For instance suppose that $P_1,\ldots,P_r$ are pairwise orthogonal projections. Then the $P_j\otimes P_j$ are also pairwise orthogonal projections. Define $M_k=\frac1k\,P_k$. Then $$ \Big\|\sum_jM_j\otimes M_j\Big\|=\Big\|\sum_j\frac1{j^2}\,P_j\otimes P_j\Big\|=1=\|M_1\otimes M_1\|=\|M_1\|^2. $$

And it doesn't look like one can improve this. In particular, the bound in the question does not hold in general. Consider $$ M_1=\begin{bmatrix} 1&0\\0&0\end{bmatrix},\qquad M_2=\begin{bmatrix} 1&1\\1&1\end{bmatrix}. $$ We have $$ \|M_1\|^2=1,\qquad\|M_2^2\|=4. $$ Also, $$ \|M_1^2+M_2^2\|=\left\|\begin{bmatrix}3&2\\2&2 \end{bmatrix} \right\|=\frac{5+\sqrt{17}}2, $$ while $$ \|M_1\otimes M_1+M_2\otimes M_2\|=\left\|\begin{bmatrix} 2&1&1&1\\1&1&1&1\\1&1&1&1\\1&1&1&1\end{bmatrix}\right\|=\frac{5+\sqrt{13}}2 $$