163 problems

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

For every Banach space $X$ and every invertible bounded linear operator $T\in\mathcal{B}(X)$, if $T$ has the shadowing property, then $T$ is generalized hyperbolic. In particular,…

For each $n\geq 1$, define $S_n$ to be the least constant such that, for every $n$-dimensional complex Banach space $X$ and every invertible operator $T\in\mathcal{L}(X)$,…

Let $X$ be a complex normed space and let $n$ satisfy $2\le n<\dim_{\mathbb C}X$. If every pair of $n$-dimensional complex subspaces of $X$ is isometric as metric spaces, then ther…

Let $N\ge 1$, $p\in[1,\infty)$, $\gamma>0$, and let $\Omega\subset\mathbb{R}^N$ be a bounded open interval if $N=1$ or a bounded Lipschitz domain if $N\ge 2$. For $\lambda>0$, defi…

Does every type $\mathrm{II}_1$ factor $M$, equipped with its normalized trace $\tau$, admit an orthonormal basis of $L^2(M,\tau)$ consisting of unitaries in $M$? Equivalently, doe…

Let $A$ be a unital operator algebra generating the $C^*$-algebra $B=C^*(A)$. If every irreducible $*$-representation $\pi:B\to B(H_\pi)$ has the unique completely positive extensi…

For $m\ge 2$ and $1<p<\infty$, define the all-dimensional best constant $c_p(m)$ by … The Tang–Zhang conjecture asserts that $c_p(m)$ equals the explicit closed-form expression pro…

For every integer $n\geq 1$ and every real $p$ satisfying $2<p<\infty$, the maximum cardinality of an equilateral set in $\ell_p^n$ is $e(\ell_p^n)=n+1$. Equivalently, every set…

Braverman–Makarychev–Makarychev–Naor conjecture. There exists $\eta>0$ such that $f_\eta$ and $g_\eta$ form a Krivine rounding scheme and

Let ${\cal H}$ be a Hilbert space, let $\Phi=(\varphi_n)$ and $\Psi=(\psi_n)$ be sequences in ${\cal H}$, and let $m=(m_n)$ be a complex sequence. The multiplier $M_{m,\Phi,\Psi}$…

Let $\mathcal{A}$ be a unital C-algebra with the invariant basis number property, and let $\mathcal{E}$ be an $n$-rank Banach module over $\mathcal{A}$. For $k\in\mathbb{N}$, write…

For $0<r_2<r_1<1$, let … be the Banach ring constructed as the non-expanding colimit of the rings $\mathbb{Z}\{(x/r)^{1/n}\}$ under the morphisms $\alpha_{n,m}$. Nuclearity conject…

Scattering convergence and continuity conjecture. Condition $$ holds for all $fin {bf L}^1({{mathbb{R}}}^d)$. Moreover, if $fin {bf L}^1({{mathbb{R}}}^d)$, then $overline S f(q)…

Let $A_\mathbb{R}(\mathbb{D})$ denote the real disc algebra, consisting of functions analytic on $\mathbb{D}$, continuous on $\overline{\mathbb{D}}$, and satisfying the relevant re…

Quantitative $R_\epsilon$-conjecture. There is a natural number $r=r(\epsilon,A,B)$ such that, for every $n\in\mathbb N$ and every such sequence, there is a partition…

$R_\epsilon$-conjecture. Every unit norm Riesz basic sequence is a finite union of $\epsilon$-Riesz basic sequences.

Asymptotic constant conjecture. There exists a constant $1\leq c\leq\sqrt e$ such that

Unitary representability conjecture. For every Tikhonov space $X$, there exists a metric space $M$ of $L^1$-type such that $F(X)$ is isomorphic to a subgroup of the group of isomet…

Let $1<p<\infty$, and let $U=\{u_k\}$ be an exponential-function family in $L^p(0,1)$, with the distance between normalized exponential-function families defined by the displayed q…

Let $\{f_i\}_{i=1}^n$ be a system of functions satisfying condition (d) with parameter $p\in[0,1/2)$. For each fixed $x\in X$, let $f_k^*(x)$ denote the decreasingly ordered values…

Montgomery-Smith–Semenov hypothesis. There exist signs $\theta_i\in\{-1,1\}$ and a constant $c_0>0$ such that

Let $w_n$ be given by equation $(15)$, let ${\mathcal H}_{\mathbf w}$ be the corresponding Hilbert space, and let $Mult({\mathcal H}_{\mathbf w})$ denote its multiplier algebra. A…

Let $A^1$ be the Bergman space, let $(A^1)^\perp$ denote its annihilator in the relevant dual space, let $C$ denote the subspace of constant functions, and let $\varphi_\beta$ and…

Coefficient ratio conjecture. Then