52 problems

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

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…

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

Let $T$ be a Kreiss bounded operator on a Hilbert space, meaning that there is a constant $C>0$ such that … for all $n\geq 1$. Shields' conjecture. Every Kreiss bounded Hilbert-spa…

Let $\mathcal{H}$ be a complex Hilbert space and let $T\in\mathcal{B}(\mathcal{H})$. Write $|T|=(T^*T)^{1/2}$ for the modulus of $T$ and $\operatorname{Re}T=\frac{1}{2}(T+T^*)$ for…

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…

Conjecture. The statement of the preceding proposition can be adjusted to cover these cases.

Let ${\cal H}$ be a Hilbert space, let $A=A^*\in L({\cal H})$, let ${\cal S}$ be a closed subspace of ${\cal H}$, and let ${\cal Q}_{\cal S}$ denote the set of projections onto…

Aldroubi–Cabrelli–Krishtal–Molter conjecture. This system is not a frame for $H$.

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…

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…

Let $X$ be a Hilbert space, let $Y$ be a Banach space, and let $\varphi\!:{\mathsf{B}}(X)\to Y$ be a bounded linear map. For an operator $Z\in{\mathsf{B}}(X)$ of generalized index…

Let $X$ be a Hilbert space, let $Y$ be a Banach space, and let $\varphi\!:{\mathsf{B}}(X)\to Y$ be a bounded linear map. For a singular operator $Z\in{\mathsf{B}}(X)$, let…

Infinite-dimensional Breuer–Major correlation summability conjecture. This condition can substitute the finite-dimensional assumption $\operatorname{dim}(\mathcal{H}_1)<\infty$ as…

Let $G$ be a gauge group, let $C$ be a Riemann surface, let $\kappa\in\kappa_c+\mathrm{i}{\mathbb R}$, and let $Z_{\kappa;\ell}$ be the partition function of the corresponding WZW…

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…

Let $H$ be a complex separable Hilbert space with $\dim H=\infty$, and let $T:H\to H$ be a bounded linear operator. A subspace $V\subset H$ is non-trivial if it is proper, closed,…

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…

Let $A$ be an operator acting on an infinite-dimensional separable Hilbert space $\mathcal{H}$. For a subspace $\mathcal{S}\subset\mathcal{H}$, let $A_{\mathcal{S}}$ denote the cor…

Let $\mathcal{H}$ be a Hilbert space and let $E(\mathcal{H})$ denote the class of operators referred to in the source. Let $T\in E(\mathcal{H})$. The Hilbert space $\mathcal{H}$ ca…