12 problems

The converse compactness assertion. If the necessary condition (e1) and condition (e2) from Theorem … remains valid throughout this remaining parameter range. The paper presents th…

Spectral radius extension conjecture. The spectral radius formulae of Theorem $$ are valid for all $\varphi\in\mathcal{S}_m$.

Let $v$ be an admissible weight, let $X$ be either $\operatorname{BMOA}_v$ or $\operatorname{VMOA}_v$, and assume $X\not\subset H^\infty$. Let $\delta_z$ denote evaluation at $z$,…

Let $v$ be an admissible weight, and let $\phi\colon\mathbb D\to\mathbb D$ be an analytic polynomial symbol. The polynomial-symbol conjecture. The composition operator … is bounded…

Let $v$ be a weight, let $\phi\colon\mathbb D\to\mathbb D$ be an analytic symbol, and let $\psi$ be an analytic function. The boundedness-assumption conjecture. The assumption in T…

Let $\mathbb D$ denote the unit disk, let $v\colon\mathbb D\to(0,\infty)$ be an increasing radial weight that is not admissible, let $\hat\phi\in\operatorname{Aut}(\mathbb D)$, and…

Let $\mathbb D$ be the unit disk, let $\mathcal{S}(\mathbb D)$ denote the holomorphic self-maps of $\mathbb D$, and for $\varphi\in\mathcal{S}(\mathbb D)$ let…

Let $\mathbb D$ be the unit disk, let $\varphi$ be an analytic self-mapping of $\mathbb D$, and let $C_{\varphi}f=f\circ\varphi$ be bounded on the Dirichlet space…

Let $\mathbb{D}$ be the open unit disk, and let $\varphi$ be an essentially linear fractional self-map of $\mathbb{D}$ with exactly one fixed point $w$ in $\overline{\mathbb{D}}$.…

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 $\mathbb{D}$.…

Numerical-range conjecture. The numerical range of $C_\varphi$ on $H^2(D)$ is the interior of the convex hull of an algebraic curve of class $p$ and degree $p^2-p$. Moreover, the r…

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…