9 problems

Let $A,B\in\mathbb{M}_n^+$ and let $p,q>0$. A matrix norm is symmetric if it is unitarily invariant. Conjecture B. For every symmetric norm, … This is presented as another conjectu…

Let $\mathbb{M}_n$ denote the space of $n$-by-$n$ matrices and $\mathbb{M}_n^+$ its positive semidefinite cone. A pair $(A,B)$ in $\mathbb{M}_n^+$ is monotone if $A=f(C)$ and…

Let the seminorms considered in Theorem 2.6 be the seminorms on $\mathbb{R}^n$ obtained from symmetric norms as defined in the paper. A seminorm $L$ is strongly Leibniz if, in addi…

Let a positive semidefinite matrix in $\mathbb{M}_{2n}^{+}$ be written in blocks of the same size, with normal off-diagonal blocks. Normal-block majorization conjecture. The inequa…

Bourin's normal-block extension of Theorem 2.5. The same inequality should hold for all normal block matrices partitioned into blocks of the same size and for all symmetric norms.…

Bourin's symmetric-norm extension of Corollary 2.2. The same inequality should hold for all symmetric norms:

Bourin's symmetric-norm extension of Corollary 1.2. The same inequality should hold with the operator norm replaced by every symmetric norm. In particular, the trace inequality (2)…

Bourin's block-matrix subadditivity conjecture. This estimate should remain true with $t^p$ replaced by every non-negative concave function $f$ on $[0,\infty)$:

Let $A$ and $B$ be normal matrices of the same size, let $Z$ be an expansive matrix, meaning $Z^*Z\ge I$, let $f:[0,\infty)\to[0,\infty)$ be non-negative and concave, and let…