49 problems

Jordan-curve conjecture. The following statements are equivalent:

Let $H^2$ be the Hardy space on the unit disk $\mathbb D$, let $f,g\in H^2$, and let $T_f$ and $T_{\overline g}$ denote the corresponding Toeplitz operators on $H^2$. For…

Let a Toeplitz operator on the Bargmann space have a symbol that is an exponential of a complex quadratic form, and let its Weyl symbol be the corresponding Weyl symbol. Berger–Cob…

Let $\mathcal{PH}_\alpha^2(\mathbb{C}^n)$ be the pluriharmonic Fock space, let $\mu$ be a positive Borel measure, and let $T_\mu$ be the associated Toeplitz operator. For $r>0$, wr…

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

Let $L_a^2(\mathbb{D})$ be the Bergman space of square-integrable analytic functions on the unit disk, and let $T_\varphi$ denote the Toeplitz operator on $L_a^2(\mathbb{D})$ with…

Hankel-product boundedness conjecture. The Hankel product $H_{\overline{f}}^{*}H_{\overline{g}}$ is bounded if and only if $D(f,g)$ is bounded.

Toeplitz-symbol convergence conjecture. The series $\widetilde{T}(z)$ converges for every

Let $X$ be an odd-dimensional manifold with the geometric data used to define the eta-type invariant, let $g$ be the corresponding gauge transformation, and let $D^{[0,1]}$ be the…

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

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…

For symbols $\varphi$ and $\psi$, let $T_{\varphi}$ and $T_{\overline{\psi}}$ be the corresponding Toeplitz operators on the Hardy space $H^2(\mathbb{T})$, and let $|\varphi|(z)$ a…

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…

Virtanen's conjecture. The Toeplitz operator $T_a$ on $H^1(\mathbb{T})$ should be Fredholm.

No-symmetry-breaking conjecture. Without assuming circular symmetry, the ball maximizes

Let $X$ be a compact complex manifold of dimension $n$, let $L$ be an ample line bundle over $X$ with a smooth positive Hermitian metric, and let $f\in L^{\infty}(X)$ be nonzero wi…

Let $(\rho_1,\rho_2)$ be a pair of weights on the unit circle, and consider the two-weighted Riesz projection … For $|w|<1$, let $P_w(\rho_1)$ denote the Poisson extension of…

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 $S$ be an operator in the Toeplitz algebra $\mathfrak{T}(L^\infty)$, and let $B_\alpha(S)$ denote the higher-order Berezin transform, generalized from…

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 $T_\Phi$ be a subnormal Toeplitz operator whose symbol $\Phi$ is matrix-valued and rational. CHKL conjecture. $T_\Phi$ is unitarily equivalent to a direct sum of a normal opera…

Let $F^p_\phi$ be a doubling Fock space over $\mathbb{C}^n$, with $n>1$, and let $T_f$ be a Toeplitz operator on this space. The paper's main Fredholm result asserts a characteriza…

Let $F^{2,m}(\mathbb{C})$ be the Fock–Sobolev space, let $\mathcal{P}$ denote the polynomials, let $K_m(z,a)$ be the reproducing kernel, and define … For symbols $f,g$, let…

Let $Omega$ be a domain with Lebesgue measure $nu$, and let $BS_{m,vartheta}(Omega)$ denote the operator class used in the source. Suppose that the commuting $d$-tuple…

Let $\mathbb D$ be the unit disk, let $A^2(\mathbb D)$ be the Bergman space, and let $T_\varphi$ be the Toeplitz operator on $A^2(\mathbb D)$ with harmonic symbol $\varphi$ on…