10 problems

Let $\mathscr{P}_{n}(\mathbb{D})$ denote the cone of positive definite $n\times n$ matrices. For $f(t)=\frac{1+t}{2}$, the associated Wasserstein mean is … A mean is linearizable w…

Infinite-dimensional extension conjecture. These results should also hold without the assumption of finite dimensionality.

Let $K_r$ denote the weighted Kubo–Mori mean and $L$ the logarithmic mean, and define … A function is positive definite if, for every finite set of real numbers $t_1,\ldots,t_n$, t…

The proper-subset conjecture. The class $GCV$ is a proper subset of $PMI$:

Let $\mathbb{P}$ denote the cone of positive definite operators, let $\mathcal{P}^1(\mathbb{P})$ be the class of probability measures on $\mathbb{P}$ with finite first moment, let…

Let $\Phi$ be a unital positive linear map, and let $A$ and $B$ be positive operators satisfying … and … for positive real numbers $m_{1}\leq M_{1}$ and $m_{2}\leq M_{2}$. Write…

Let $A$ and $B$ be positive matrices, let $0\leq t\leq 1$, and let $|||\cdot|||$ be any unitarily invariant norm. Hayajneh–Kittaneh conjecture. One has … The conjecture concerns co…

Let $A$ and $B$ be positive operators and let $q\in\mathbb{R}$. Complementary McCarthy conjecture. … for $-2<q<0$, while the reversed inequality should hold for $3\leq q$. The disp…

Let $A$ and $B$ be positive operators, let $t\in[0,1]$, and for $p\geq 1$ define the weighted $p$-power mean by … The results that weighted $p$-power means are distance-monotonic a…

Uniqueness conjecture for the non-commutative arithmetic-geometric mean inequality. There is no other non-commutative analogue of $a^{r}b^{1-r}$ satisfying the arithmetic-geometric…