55 problems

Lee's conjecture. The optimal constant for the Schatten $2$-norm is

Let $\mathcal{H}$ be a Hilbert space, let $A$ satisfy $0<mI\leq A\leq MI$, and let $X,Y$ be partial isometries on $\mathcal{H}$ whose final spaces are orthogonal. For every 2-posit…

Let $k$ be a real number with $0\leq k\leq 2$. For a natural number $n$, let $\lambda _1,\dots,\lambda _n$ be nonzero numbers satisfying … for all $i,j=1,\dots,n$. Positivity conje…

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…

Sharp arithmetic symmetric modulus conjecture. For every $X,Y\in\mathbb{M}_n$, there exist unitaries $U,V$ such that

Universal-constant Thompson domination. There should exist a universal constant $C$ such that, for every $X,Y\in\mathbb{M}_n$, there are unitaries $U,V$ satisfying

Let $A_1,\ldots,A_n\in S_p$, where $S_p$ is the Schatten $p$-class. For $1<p<\infty$, let $q$ be the conjugate exponent of $p$, so that $\frac1p+\frac1q=1$. Audenaert–Kittaneh conj…

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 $A,B\in\mathbb{H}_n$ 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…

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…

Optimal-bound conjecture. The optimal state-dependent constant is

Let $A$ and $B$ be observables of a quantum system in state $\rho$. Write $V_\rho(A)$ and $V_\rho(B)$ for their variances, $[A,B]$ for their commutator, and $\|[A,B]\|_\rho$ for th…

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…

Optimality conjecture. The constant $k/4$ is optimal for every odd integer $k>1$, meaning that it cannot be replaced by any smaller constant. The source proves optimality for even…

Let $S,T,R$ be contractions in $\mathbb{M}_n$, and let $I$ denote the identity matrix. The inequality … holds. Optimality conjecture. The constant $3/4$ is optimal, meaning that it…