24 problems

Let $A,B\in\mathbb{H}_n$ be Hermitian matrices, and let $H$ and $R_k$ denote the operators defined in the paper from the coefficients of $e^{(A+B)t}$ and $e^{At}e^{Bt}$, respective…

Lin's conjecture. There exists a unitary matrix $U\in M_n$ such that

For $Z\in\mathbb{M}_d$, let $|Z|_{\mathrm{qsym}}$ denote the quasisymmetric modulus used in Zhang's conjecture. For $A,B\in\mathbb{M}_d$, there are unitaries $U,V\in\mathbb{M}_d$.…

Maximal symmetric modulus conjecture. For every dimension $d$, one may choose $c_{\vee}(d)=1$. The assertion would settle the corresponding finite-dimensional question in its stron…

Let $S_p$ denote the Schatten $p$-class, and let $A,B,C\in S_p$. For $p>0$, write $\\|\cdot\\|_p$ for the Schatten $p$-quasi-norm. Sharp three-operator Clarkson–McCarthy conjecture…

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…

Kernel equality conjecture. For every such tuple,

Let $\mathscr{H}$ be a Hilbert space, let $\mathbb{B}_p(\mathscr{H})$ denote the Schatten $p$-class, and define $\omega_j=e^{ji\pi/n}$. Let…

Let $\mathscr{H}$ be a Hilbert space and let $\mathbb{B}_p(\mathscr{H})$ denote the Schatten $p$-class on $\mathscr{H}$. For $A,B\in\mathbb{B}_p(\mathscr{H})$, the noncommutative C…

Let $A_1,\ldots,A_n\in\mathbb{M}_m$, and let $|A|$ denote the positive operator $(A^*A)^{1/2}$. Zhang's conjecture. For $p>2$, there exist $\frac{n(n-1)}{2}+1$ unitary matrices $U$…

Audenaert–Kittenah conjecture. For $1\le p\le 2$,

Bourin–Lee's multivariable conjecture. The unitary matrices can be chosen so that

Audenaert–Kittaneh's conjecture. For $1<p\le 2$,

Let $A$, $B$, and $X$ be operators in $B({\mathcal H})$. Assume that $A,B\geq 0$ and that $X$ is compact with finite unitarily invariant norm…

Let $A$, $B$, and $X$ be operators in $B({\mathcal H})$. Assume that $A,B\geq 0$ are compact with finite unitarily invariant norm…

All-order trace derivative positivity conjecture. The function $t\mapsto\operatorname{tr} f(A+tB)$ has non-negative $k$th derivative.

Al-Rashed–Zegarliński's quadratic-form conjecture. For every completely positive trace-preserving map $\phi$ on $B(H)$ and all $A,B,D\in B(H)$,

Al-Rashed–Zegarliński's conjecture. If $p\in[1,2)$, then for every $X\in B(H)$ and every unital completely positive trace-preserving map $\phi$ on $B(H)$,

Let $A$ and $B$ be positive matrices, let $f$ be a function on $[0,\infty)$ with $f(0)=0$, and let $|||\cdot|||$ be an arbitrary unitarily invariant norm. Ando-type conjecture. If…

Riesz projection contractivity conjecture. The operator $P$ is a contraction from $L^r(\mathbb{T})$ to $H^q$. In addition, for every $f\in L^1(\mathbb{T})$,

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…

Trace-norm commutator conjecture. For certain functions $f$ still to be determined, there exists a constant $c$ independent of $d$ such that

Let $0<m\le A\le M$, and let $X,Y$ be partial isometries on a Hilbert space $\mathcal{H}$ whose final spaces are orthogonal to each other. Let $\Phi$ be a $2$-positive linear map.…

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…