195 problems

Let $\alpha$ be a continuous $\|\cdot\|_1$-dominating normalized gauge norm, and let $H^{\alpha}$ and $M_{\alpha}(z^n)$ be the associated Hardy space and invariant subspace. Decomp…

Near-isometric duality conjecture. For every $n\times n$ matrix $A$ with $n>2$,

Is every Banach space $X$ with the renorming-stable ball-covering property separable? More precisely, if for every norm $\lVert\cdot\rVert'$ equivalent to the given norm on $X$, th…

For every integer $N\ge 1$ and every linear subspace $V\subset \ell_\infty^N$ with absolute projection constant $\lambda(V)<\frac{4}{3}$, one has …

For every separable Hilbert space, every $p\in[0,1]$, and all positive trace-class operators $A\leq B$ satisfying $\operatorname{Tr}A=p$ and $\operatorname{Tr}B=1$, prove the sharp…

For each dimension $N\ge 2$ and higher order $m\ge 2$, determine the sharp constants and all extremal divergence-free vector fields in the corresponding higher-order Heisenberg unc…

Given a bounded vertex-weight family $\mu=(\mu_{v_{i,j}})_{(i,j)\in\mathbb{N}\times\mathbb{N}}$ and the associated backward shift $B_{\mu}$ on the sequence space of the infinite-qu…

For integers $n\ge 2$ and $1\le p\le\infty$, let $\lambda_3(n,k,p,q)$ be the optimal constant in the inequality $\|u^{(k)}\|_{L^q(0,1)}\le \lambda_3(n,k,p,q)\|u^{(n)}\|_{L^p(0,1)}$…

For every compact group $G$, the central Fourier algebra $\operatorname{ZA}(G)$ is amenable if and only if $G$ is virtually abelian, i.e. $G$ has an abelian subgroup of finite inde…

Let $X$ be a real Banach space with $\dim X>1$. The Daugavet property is the condition that $\|I+S\|=1+\|S\|$ for every rank-one operator $S\colon X\to X$. The square-Daugavet prob…

For every dimension $d\in\mathbb{N}$ and smoothing parameter $\beta>0$, determine the complete spectral decomposition of the Henze–Wagner covariance-kernel integral operator…

Let $\rho^{A_1\cdots A_m}$ be an $m$-party quantum state and let $\varepsilon>0$. Writing $\rho^K$ for the marginal on every nonempty subset $K\subseteq\{1,\ldots,m\}$, the conject…

For every pointed metric space $(M,p)$, the Lipschitz-free space $\mathcal{F}(M)$ is the strongly unique isometric predual of $\operatorname{Lip}_0(M)$; that is,…

For each $q\ge 2$, does there exist a universal constant $C_q<\infty$ such that every bounded linear operator $T:X\to Y$ between Banach spaces satisfies … where…

Fix $0<\alpha,r<\infty$. For every $0<p,q\leq\infty$, determine whether, for every $f\in C^1([-1,1])$ satisfying $f(0)=0$ and $|f'(t)|\lesssim |t|^\alpha$, and every real sequence…

Let $C$ be a nonempty closed, bounded, convex subset of a Banach space, let $f:C\to C$ be continuous, and suppose that $f^n$ is compact for some integer $n\ge 1$. Then $f$ has a fi…

Let $E$ be a Banach space, and let $(y_n^*)_{n=1}^{\infty}\subset S_{E^*}$ be a normal sequence, meaning that $y_n^*(x)\to 0$ for every $x\in E$. Must $(y_n^*)$ have a subsequence…

Let $X$ be a Banach space, and let $A,B:X\rightrightarrows X^*$ be maximally monotone operators. If $\operatorname{dom}A\cap\operatorname{int}(\operatorname{dom}B)\neq\varnothing$,…

Let $\mathbb{T}$ be the unit circle, let $P_+$ denote the Riesz projection onto the nonnegative Fourier modes, and let $1<q<\infty$. The conjecture asserts the sharp contractive es…

Let $\Omega\subset\mathbb{R}^d$ be a bounded $\mathrm{C}^{1,\alpha}$-domain, and let $A_{1,\sigma,n}$ denote the Stokes operator with no-slip boundary conditions on…

For $1\le p<\infty$, even $m\ge 4$, and $n\in\mathbb N$, let $A_p(n,m)$ be the least constant such that every function $f:\mathbb Z_m^n\to L_1$ satisfies … Determine the optimal or…

There exists a constant $\varepsilon>0$ such that, for every measurable function $f:\mathbb{R}\to\mathbb{R}^{+}$ for which the relevant Gowers norms are finite and nonzero,…

Let $X$ be a Banach space over $\mathbb{R}$ or $\mathbb{C}$ having an unconditional Schauder basis, and let $Y$ be a complemented closed subspace of $X$, meaning that there exists…

Let $\Omega_n=\{-1,1\}^n$ with normalized uniform measure, and let $\nabla$ and $\Delta$ denote the Walsh gradient and Walsh Laplacian, respectively. The conjecture asks whether, f…

Given Schur-class functions $b_1,b_2$ on the unit disk $\mathbb{D}$ and an analytic self-map $\varphi:\mathbb{D}\to\mathbb{D}$, characterize when the composition operator…