8 problems

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 $\mathbb{H}=H^2(\mathbb{C},B^n(0,1),d\mu)$ be the Hardy space of complex-valued functions on the $n$-dimensional unit ball, whose boundary is the unit sphere $S^{2n-1}$. A stan…

Let $\mathrm B_m$ and $\mathrm B_\alpha$ denote the matrix balls, and let $\mathop {\mathrm {Inn}}\nolimits(m,\alpha)$ be the space of all rational inner maps…

Approximation conjecture. Any solution $u$ of this equation such that $u\leq u_{I',\infty}$ can be approximated uniformly on compact subsets by solutions $u_k$ that are bounded abo…

Let $A^2_1$ be the weighted Bergman space of holomorphic functions on the unit disk with norm … For $H\in A^2_1$, write … the smallest closed subspace of $A^2_1$ containing $H$ and…

Characterization conjecture. $A\in\mathcal{T}(\theta)$ if and only if $A_\alpha$ is $C_\alpha$-symmetric for every nonconstant inner function $\alpha$ dividing $\theta$.

Let $\Theta$ be a $d\times d$ matrix-valued inner function on the bidisk, and let $S_{z_1}$ denote the associated compressed shift operator. Write $\deg_1\Theta$ and…

One-component characterization conjecture. The equivalent conditions of Theorem 3 are fulfilled if and only if $\theta$ is one-component.