17 problems

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 $\mathcal{C}$ 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…

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 $R\geq 2$ 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 $p$ 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…

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 $X\in\mathbb{M}_n$. 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…

General independent-entry matrix norm conjecture. Then $$

Rademacher matrix norm conjecture. We have

Let $A$ be a random matrix in $\mathbb R^{m\times k}$ with independently and identically normally distributed entries, and let $L$ be the synthesis operator appearing in the formul…

Let $A$ be an $n \times n$ random matrix with i.i.d. mean-zero entries satisfying … For a suitable $L_m$-norm, let $\tilde{A}$ be obtained from $A$ by zeroing out every row and col…

The $p\to q$ norm conjecture. For any $\kappa\ge1$ and any $p,q\ge1$, every quantum state $\hat\rho$ satisfies

Let $P$ be the Bergman projection on the unit disk. For $1<p<\infty$, let $q:=p/(p-1)$ be the conjugate exponent and let $\|P\|_p$ denote the operator norm of $P$ on…

Sharp Bergman projection norm conjecture. The upper quantity is expected to attain the lower bound:

Let $n$ be a positive integer, let $\mathbf{F}$ be the normalized $n\times n$ DFT matrix, and let $\mathsfsl{P}_\delta$ and $\mathsfsl{P}_\delta'$ denote the coordinate projections…