14 problems

Foci conjecture. If $\operatorname{cl}(W(T))$ is an elliptical disk, then its foci are contained in $\{0,1,-1\}$.

Clouâtre–Ostermann–Ransford conjecture. (i) If $\|\Lambda\|\leq 2$, then $\|\theta\|\leq 2$. (ii) If $\alpha$ is a complete contraction, $\theta$ is completely bounded, and…

Gau–Wang–Wu conjecture. If the numerical range $W(A)$ is a circular disc centered at the origin, then $A$ is rotationally invariant.

Let $t\in[0,1)$, and let $\Theta_t$ be of the unicritical form referred to in the source. Let $W(\Theta_t)$ denote the associated numerical range. Pseudohyperbolic disk conjecture.…

Level set Crouzeix conjecture. One has

Let $X\in\mathbb{M}_n$. Say that $X$ has the universal block-matrix inequality if, for every positive block matrix with $X$ as its off-diagonal block, … Essentially Hermitian conje…

Let $\mathfrak{R}$ be a type-$\mathrm{II}_{\infty}$ or type-$\mathrm{III}$ factor acting on a separable Hilbert space, and let $W_e^{\mathfrak{R}}(A)$ denote its essential numerica…

Let $S(a_1,\dots,a_n)$ denote the $n\times n$ cyclic shift matrix with weights $a_1,\dots,a_n$. For $n\geq 2$, let $0\leq a_1\leq\cdots\leq a_n$ and set … For a permutation…

Let $F_B\in\mathbb{C}[t,x,y]^{C_n}_d$ be the homogeneous polynomial associated with a matrix $B\in\mathbb{C}^{d\times d}$, and let $\mathcal{W}_k(B)$ denote its $k$-th numerical ra…

Let $B\in \mathbb{C}^{d \times d}$, and suppose that its numerical range $\mathcal{W}(B)$ is invariant under multiplication by $e^{2\pi i/n}$. Let $\mathcal{C}_{\mathbb{C}}(n,d)$ d…

Hollowisation conjecture. There exists an orthogonal matrix $V\in\mathbb{R}^{n\times n}$ such that $V^TAV$ is hollow and $VBV^T$ is almost hollow.

The numerical-range conjecture.

Borcea's normal-matrix spectral-distance conjecture.

Suppose $T\in {\mathcal L}({\mathcal H})$ is a contraction with … Let ${\mathcal U}$ be the set of unitary $N$-dilations of $T$ to ${\mathcal H}\oplus \mathbb{C}^N$, and define…