28 problems

Let $\mathbb D$ be the unit disc and $H^2(\mathbb D)$ its Hardy space. For an analytic self-map $\varphi$ of $\mathbb D$, define the composition operator $C_\varphi$ by … Let…

Let $1\leq p<\infty$, let $\varphi$ and $\psi$ be analytic self-maps of the unit disk $\mathbb{D}$, and let $C_\varphi f=f\circ\varphi$ and $C_\psi f=f\circ\psi$ denote the associa…

Let $U^n$ be the unit polydisk, let $p,q\in(0,1)$, and let $\phi$ be a holomorphic self-map of $U^n$. Write $C_\phi f=f\circ\phi$ for the composition operator, and let…

Let $m>1$, let $C_{\psi,\phi}$ be a bounded weighted composition operator on the weighted Fock space under consideration, and suppose that the affine symbol has the form…

Triebel's composition conjecture. If $s>1+\frac{1}{p}$, this map is bounded if and only if

Infinite-complement conjecture. The set $T\setminus\varphi(T)$ should be infinite.

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…

Automorphism conjecture. Every algebra automorphism of $\mathcal{A}$ must be a composition operator induced by some disc automorphism.

Weak compactness conjecture. If $p^{-}=1$, then $T_{\varphi}$ is weakly compact on $L^{p(\cdot)}([0,1])$ if and only if

Let $(C_t)_{t\geq 0}$ be a composition semigroup on the weighted Bergman space $A^p_\omega$, with infinitesimal generator $\Gamma$, symbols $\varphi_t$, and Denjoy–Wolff point $b$.…

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…

Minimality conjecture. Under this assumption, $K_g$ is minimal if and only if $g$ is a $C_{\phi_a}$-eigenvector. This conjecture concerns the classification of minimal invariant su…

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 $B_p$ be the analytic Besov space with $p>2$, and let $C_{\varphi}$ denote the composition operator induced by an analytic self-map $\varphi$ of the disk, defined by…

Let $\Omega$ be a bounded set, and let $H^2(\Omega)$ denote the Hardy-Smirnov space on $\Omega$. For an analytic self-map $\Phi$ of $\Omega$, write $C_{\Phi}$ for the composition o…

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 $A:= \{\mathcal H(\beta,\Lambda): \beta_* = \infty\}$, and let $\{\mathcal H(E,\beta_S)\}$ be a proper subset of $A$. Consider bounded composition operators on these spaces and…

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 $\mathbb{D}$.…

Conjecture on composition operators and weak quasiconformal mappings. The language of composition operators and its geometric representations as weak quasiconformal mappings should…

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 $1\leq p<\infty$, let $\varphi$ and $\psi$ be analytic self-maps of the unit disk $\mathbb{D}$, and let $C_\varphi f=f\circ\varphi$ and $C_\psi f=f\circ\psi$ act on $H^p$. Let…

Non-optimality conjecture. The exponent $(2^\alpha-1)/d$ is not optimal for describing the relevant composition-operator embedding.