45 problems

Let $\mathcal{H}_A$ and $\mathcal{H}_B$ be finite-dimensional Hilbert spaces, and let $\mathcal{T}_A$ and $\mathcal{T}_B$ be super-operators on $\mathcal{H}_A$ and $\mathcal{H}_B$,…

Let $X\in\mathbb{M}_n$. Say that $X$ has the universal block-matrix inequality if, for every positive block matrix with $X$ as its off-diagonal block, … Essentially Hermitian conje…

DGS conjecture. The number $c_p$ is the operator norm of $M$ on $L^p(\mathbb{R})$, that is,

Let $\mathcal{T}$ range over triangulations of bounded polygonal domains in $\mathbb{R}^2$, let $\mathcal{T}$ have at most $N$ triangles, and let $P_{V(\mathcal{T})}$ denote the…

Let $R\geq 2$ and let … where $t_1,\ldots,t_R$ are distinct points on ${\bf T}$, $0<\operatorname{Re}\alpha_r<1/2$, $-1/2<\operatorname{Re}\beta_r\leq 1/2$, and…

Let ${\mathcal A}_{2m}^\mu$ be the reconstruction operator in Algorithm N/2, acting from $C(B^2)$ to $C(B^2)$, where $\mu$ is the attenuation parameter and $m$ tends to infinity. W…

Let the Hilbert–Schmidt norm on matrices be the distance under consideration, and let completely positive trace-preserving maps act on the space of matrices. Vedral et al.'s conjec…

For fixed $N$, let $B_{N-1,K}$ and $W_N^{\rm opt}$ denote the two-branch and standard dyadic Walsh--Hadamard truncation matrices described in the paper, with $K=0,\dots,N$. Two-bra…

Let $\mathcal{C}$ be a set of columns of a dyadic-truncated Walsh--Hadamard matrix $W_\Phi$ with exactly one node, such that each branch is complete above the node. Let…

The optimal distance conjecture. For every nonnegative, nonincreasing function $f$,

Let $M_{m,n}(\mathbb{C})$ denote the space of complex $m\times n$ matrices. For a matrix, write $\|\cdot\|_F$ for the Frobenius norm, $\|\cdot\|_2$ for the spectral norm, and…

Let $p$ and $q$ satisfy $p\le 2\le q$, and let $A=(a_{ij})$ be a deterministic matrix. Let $(\varepsilon_{ij})$ be independent symmetric Bernoulli random variables, taking the valu…

Let $p$ and $q$ satisfy $p\le 2\le q$, and let $A=(a_{ij})_{i\le m,j\le n}$ be a deterministic $m\times n$ matrix. Write $p^*$ for the Hölder conjugate of $p$, and let…

Let $T_n$ be the Cauchy-Toeplitz matrix … For $1\leq p,q\leq\infty$, let $\|T_n\|_{p,q}$ denote its $\ell_{p,q}$ norm. Bozkurt's conjecture. The inequalities … and … are valid. The…

Let $M$ be the matrix associated with a one-dimensional hyperbolic boundary control system, let $\mathbf{U}(s)$ be the corresponding matrix-valued function, let $n$ be the system d…

Adamczak et al.'s dimension-free conjecture. The logarithmic factor in the available two-sided estimate for $\|A\|_{r\rightarrow p}$ should be removable, yielding a dimension-free…

Let $\Omega_n$ be the Birkhoff polytope, $J_n$ the matrix whose entries are all $1/n$, and let $R_p(\Omega_n)$ denote the Chebyshev radius of $\Omega_n$ with respect to the operato…

Non-tensorial Weibull norm conjecture. The expected operator norm is comparable, with constants depending on $p$ and $q$, to $D_1+D_2+D_{3,r}$.

Let $A_m^{\text{Had}}$ denote the normalized Hadamard matrix associated with the parameter $m$, and let $\lVert A_m^{\text{Had}}\rVert_{\infty,1}$ denote its operator norm from…

For $p\in(1,\infty)$, let the Riesz–Titchmarsh discrete Hilbert transform be the discrete Hilbert transform on $\ell^p(\mathds{Z})$, and let the classical Hilbert transform act on…

Sharpness conjecture. One has

Let $A=(a_{ij})_{1\leq i\leq m,1\leq j\leq n}$ be a real matrix, let $(g_{ij})$ be independent standard Gaussian random variables, and set $G_A=(a_{ij}g_{ij})$. For…

General independent-entry matrix norm conjecture. Then $$

Rademacher matrix norm conjecture. We have

Let $A_0$ and $B_r$ be local operators at sites separated by distance $r$ in a one-dimensional spin chain with power-law interactions of exponent $\alpha$, and let…