20 problems

Let $c_n(X)$ denote the $n$th linear polarization constant of a normed space $X$, and let ${\bf R}^n$ be the real Euclidean space. Révész–Sarantopoulos conjecture. … This conjectur…

Let $G$ be a locally compact, compactly generated group. A Hilbert length function with linear growth on $G$ is a Hilbert length function whose growth is linear. Linear-growth Hilb…

Bergelson–McCutcheon IP-limit positivity conjecture. For every $f\in\mathcal H$,

Let $\fX\in 3\mathsf{Vect}$ be an $O(3)$-fixed point. For every $F\colon 2\mathsf{Vect}\to\fX$, suppose that $F$ is equipped with an $O(2)$-fixed point structure, and that…

Let $\mathcal{F}$ be a family in a separable Hilbert space $\mathcal{H}$. The family is complete if its closed linear span is $\mathcal{H}$, and it is a weighted lower semi frame i…

Let $(\mathcal{F}_{\mathbb{M}}^{\mathbb{M}},{\Greekmath 011A})$ be a Hilbert function space. Assume that $\mathcal{E}=\{e_i\}_{i\in I}\subset \mathcal{F}_{\mathbb{M}}^{\mathbb{M}}$…

Let $H^2(\mathbb{D}^2)$ be the Hardy space on the bidisk, and for a submodule $M$ define … where $R_z$ and $R_w$ are multiplication by the coordinate functions on $M$. Let…

Let $H^2(\mathbb{D}^2)$ be the Hardy space on the bidisk, and let $M$ be a submodule. Define … where $R_z$ and $R_w$ are multiplication by the coordinate functions on $M$. The subm…

Hilbert-space characterization conjecture. The following conditions are equivalent: $\mathbb{X}$ is a Hilbert space; and a nonempty set $A\subset S_{\mathbb{X}}$ is a minimal…

Paszkiewicz's conjecture. The sequence $(S_n)$ converges strongly.

Let $E$ be a closed subset of the unit circle $\mathbb{T}$ such that $|E|=0$. Let $\mathcal{F}$ and $\mathcal{F}'$ be separable Hilbert spaces, and let…

Let $E$ be a closed subset of the unit circle $\mathbb{T}$ of Lebesgue measure zero. A Hilbert-space contraction is a contraction acting on a Hilbert space, and $T$ is unitary when…

Uniform separation conjecture. If $M$ is bounded and uniformly separated, then $M$ is $ell^1$-bounded. This is posed as an open question in the paper; no resolution is supplied the…

Weak conjecture. Every pre-quasi Gelfand triple of Hilbert spaces is a quasi Gelfand triple.

Feichtinger conjecture. The sequence $\{\tau_n\}_n$ can be partitioned into a finite union of Riesz sequences for $\mathcal{H}$. This conjecture was proved by Marcus, Spielman and…

Existence conjecture. There exists a non-normal EP operator $T$ on $\mathcal H$ such that

Let $(x_*,0,\lambda_*)$ be a simple solution of the eigenvalue problem, meaning that the first three hypotheses stated in the paper hold. Suppose that $G$ and $H$ are separable, an…

Aliasing-norm conjecture. The formula in Proposition should hold for the consistent sampling corresponding to the oblique decomposition

Let $d\geq 2$, let $x_1,\ldots,x_d$ be unit vectors in $\mathbb{R}^d$, and let $v$ be a unit vector in $\mathbb{R}^d$. The $d$th linear polarization constant $c_d(\mathcal{H})$ is…

Fong–Tsui conjecture. If