18 problems

Essential positivity conjecture. The operator $T_f$ is essentially positive if and only if

Let $\mathcal M(\Sigma,t)$ be the moduli space associated with a surface $\Sigma$ and parameter $t$. In the higher-genus setting, model $\mathcal M(\Sigma,t)$ by a toric manifold,…

Curto–Hwang–Lee conjecture. There exists a finite Blaschke–Potapov product $B\in\mathcal{E}(\Phi)$ such that

Perälä–Virtanen conjecture. The operator $T_f$ is essentially positive if and only if

Jordan-curve conjecture. The following statements are equivalent:

Let $H^2(\mathbb C^n,d\mu)$ be the Bargmann–Fock space, let $g$ be measurable, and let $k_a$ denote the normalized reproducing kernel at $a\in\mathbb C^n$. For $t>0$, define the he…

Let … of $^n$, and let $1<pbackslashle 2$. Let $A^p$ denote the Bergman space, let $T$ be the Toeplitz operator induced by … be its Berezin transform, and let $K(,)$ be the Bergma…

Let $E_1$ and the relevant Hardy-space-valued symbols be as in Theorem 1, with $T_{\Phi}$ and $T_{\Theta}$ the associated Toeplitz operators. Define … Here a half-space is a closed…

Let $m\neq 0$, let $f,g,h,k\in\mathcal{A}_1$, and define … Here $T_u$ and $T_v$ are the Toeplitz operators with symbols $u$ and $v$ on the Fock-Sobolev space $F^{2,m}$. Zero-produc…

Let $d$ and $N$ be integers, and let $U,V$ be open subsets of $(\mathbb{S}^2)^d$. Write $\operatorname{dist}(U,V)$ for their distance and let $S_N$ denote the Toeplitz operator app…

Let $t>0$ and let $f$ be a symbol for which the Toeplitz operator $T_f^t$ and its Berezin transform $tilde{f}^{(t/2)}$ are defined on the $t$-Fock space. Berger–Coburn conjecture.…

Scalar-limit conjecture.

Let $n$ be the fixed positive integer defining the polyanalytic Bargmann–Segal–Fock setting. A bounded square-root-oscillating sequence is a bounded sequence with the square-root o…

Let $H^2$ be the Hardy space, let $\boldsymbol{D}$ be the unit disk, and let $C_{\varphi}$ denote the composition operator induced by a holomorphic self-map $\varphi$ of…

Sarason's conjecture. The operator $\mathcal{T}_f\mathcal{T}_g^*$ is bounded on $H^2(\mathbb{T})$ if and only if

Sarason's conjecture. The operator $T_fT_g^*$ is bounded on $A^2(\mathbb{D})$ if and only if

Let $T_\phi$ be a scalar Toeplitz operator on the Hardy space whose self-commutator $[T_\phi^*,T_\phi]$ has finite rank. Normal-or-analytic conjecture. If $T_\phi$ is subnormal, th…

Let $\hat{J}$ and $\hat{H}$ be the semiclassical Toeplitz operators associated with a semitoric system near a focus-focus singularity. The singular Bohr–Sommerfeld formula describe…