14 problems

Let $T$ be an $N\times N$ normal matrix, and let $k\geq 1$ be a fixed positive integer. Write $\Lambda_k(T)$ for the rank-$k$ numerical range of $T$, and let $\Gamma$ range over al…

Choi–Kribs–Życzkowski's convexity conjecture. Higher-rank numerical ranges $\Lambda_k(A)$ are always convex. This conjecture was proved by Woerdeman in 2008, so the claim is solved…

Let $A$ be an $n$-by-$n$ nilpotent matrix, and let $W(A)$ denote its numerical range. A flat portion is a line segment contained in the boundary of $W(A)$. Gau–Wu conjecture. The b…

Extended local-permutation conjecture. Every element of $E(\mathfrak{t}_{\mathrm{loc}})$ is similar to a block-shift matrix via an element of $\Pi_{\mathrm{loc}}^{\mathrm{ex}}$.

Irrational-angle counterexample conjecture. It is impossible to find a counterexample satisfying $\varphi_0\notin 2\pi\mathbb{Q}$.

Factorization conjecture. If $\theta$ does not factor in this way, then $W(S_\theta^1)$ and $W(S_\theta^2)$ are open. If $\theta$ does factor in this way, then $W(S_\theta^1)$ and…

Higher-degree similarity conjecture. A similar argument should work in theory, though perhaps not in practice, for the higher values $n\ge6$. For $n=4$ and $n=5$, the argument yiel…

Let $\Theta$ be a unicritical finite Blaschke product, meaning that for some $z_0\in\mathbb{D}$, $n\ge1$, and $\lambda\in\mathbb{T}$, … Let $S_\Theta$ be the associated compression…

Let $T\in M_n$ be a normal matrix, let $1\leq k\leq n$, and let $\lambda_1,\ldots,\lambda_n$ be the eigenvalues of $T$. For a finite set of complex numbers, write…

Unital uniform-algebra conjecture. If $\|\theta_\alpha\|\leq 1$, then

Let $T$ be a normal operator on an $n$-dimensional complex inner-product space, with eigenvalues $\lambda_1,\ldots,\lambda_n$. For a positive integer $k$, define the higher-rank…

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 $\Theta$ be the two-variable inner function under consideration, and let $S^1_{\Theta}$ denote the corresponding compression of the shift. Write $\theta$ and $\theta_1$ for the…

Given $S\subseteq \mathbb{C}$, let $\operatorname{conv} S$ denote the smallest convex subset of $\mathbb{C}$ containing $S$. Let $A\in M_n$ be a normal matrix with eigenvalues…