19 problems

Let $I\subseteq\mathbb{C}[\mathbf{z}]$ be an ideal, let $\mathcal{Z}$ be its zero variety, and let $\mathcal{S}$ be the submodule obtained by taking the closure of $I$ in…

Let $\Omega$ be a bounded, strongly pseudoconvex domain in $\mathbb{C}^m$, and consider submodules of $B^2(\Omega)\otimes\mathbb{C}^k$. Finite-multiplicity p-reductivity conjecture…

Let $\Omega$ be a bounded, strongly pseudoconvex domain in $\mathbb{C}^m$ with smooth boundary, let $B^2(\Omega)$ be its Bergman space, and let $\mathcal{S}\subseteq B^2(\Omega)$ b…

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…

Let $\mathbb{B}_n$ be the Bergman ball, let $p\in(0,\infty)$ and $\alpha>n$, and let $A^p_\alpha(\mathbb{B}_n)$ denote the weighted Bergman space of holomorphic functions with norm…

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

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…

Analytic Bergman-space conjecture. The statement of the smooth affine algebraic-curve theorem holds for $M$: for every open subset $U\subseteq M$, the Bergman space $A^2(U,\sigma)$…

Tight fitting conjecture. (a) The embedding $A^p_\alpha\subset A^q_\beta$ is a tight fitting if and only if

Let ${\mathbb B}_n$ be the unit ball in $\mathbb C^n$, with Lebesgue measure $dv(z)$, and define the hyperbolic measure by … For $0<p<\infty$ and $\alpha>n$, let $A^p_\alpha$ be th…

Let $M$ be a projective manifold, let $L$ be a very ample line bundle on $M$, and for each $k\in\mathbb{N}$ let $\mathcal{B}_{M,L^k}$ be the corresponding Bergman space with Calabi…

Holomorphic equivalence conjecture. If there is a linear isometry between $A^p(D_1)$ and $A^p(D_2)$, then $D_1$ and $D_2$ are holomorphically equivalent.

Let $A^2_1$ be the Bergman space … A closed subspace of $A^2_1$ is singly generated if it is the minimal closed invariant subspace containing one function, where invariance is with…

Let $\Omega\subseteq\mathbb{C}^m$ be a bounded strongly pseudoconvex domain with smooth boundary, let $A$ be the algebra of analytic functions under consideration, and let $I$ be a…

Let $D$ be a Stein domain in $\mathbb C^n$, and let its Bergman space be the space of holomorphic functions on $D$ that belong to $L^2(D)$ with respect to Lebesgue measure. Wiegeri…

Let $\mathbb D$ be the unit disk and let $L^1(\mathbb D)$ denote the space of integrable functions on $\mathbb D$ with respect to normalized planar Lebesgue measure. For a positive…

Let $0<p,q<\infty$, let $\omega\in\widehat{\mathcal{D}}$, and let $\nu$ be a positive Borel measure on $\mathbb{D}$. Let $\varphi$ and $\psi$ be analytic self-maps of $\mathbb{D}$.…

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

Sharp Bergman projection norm conjecture. The upper quantity is expected to attain the lower bound: