6 problems

Let $Q^{(i)}=[Q^{(i)}_{(jk)}]_{j,k=1}^2$ be arbitrary positive semidefinite $2\times2$ block matrices, and let $q^{(i)}_k$ be arbitrary non-negative numbers. For a matrix $A$, writ…

Let $M_n(\mathbb C)=\mathbb C^{n\times n}$, and for $A\in M_n(\mathbb C)$ write $|A|=(A^*A)^{1/2}$. For $p>1$ and $m\geq2$, let $c_p(m)$ be the smallest number such that … for all…

Let $A,B\in M_{n\times n}(\mathbb{C})$, and let $\sigma_\uparrow(A)$ denote the singular-value vector of $A$ arranged in increasing order, while $\sigma_\downarrow(B)$ is arranged…

Let $A,B\in M_{n\times n}(\mathbb{C})$, and let $\sigma_\downarrow(A)$ and $\sigma_\downarrow(B)$ denote their singular-value vectors arranged in decreasing order. For…

Let $A_0,A_1,\ldots,A_{n-1}$ be operators on a separable Hilbert space, and let $p,q$ satisfy $1/p+1/q=1$. Operator Hanner-type conjecture. For $2\leq p<\infty$, … whereas for…

Schatten-norm commutator conjecture. In the restricted case $p=q$,