39 problems

Halmos's conjecture. For every contraction $T\in B(H)$,

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

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…

Choi's convexity conjecture. The set $\Lambda_k(T)$ is convex. Equivalently, the remaining problem is to show that $0\in\Lambda_k(T')$ for every such pair of matrices $X,Y$.

Converse inclusion conjecture. The converse inclusion in the identity for $\Lambda_k(\sigma)$ stated as Eq. (normalid) should hold for arbitrary normal operators $\sigma$.

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…

Choi–Giesinger–Holbrook–Kribs conjecture.

Let $A\in \mathbb{C}^{n\times n}$ have eigenvalues $\lambda_1,\dots,\lambda_n$, and let $p(A)$ denote the matrix expression used in the determinant identity. Let $t_1(A)$ be the qu…

Crouzeix's conjecture. $K_{\mathrm{bfd}}=2$.

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.…

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…

Ridge's conjecture. If all entries of $a$ are non-zero, then $W(T_a)$ is open.

Let ${\bf A}\in\mathbb{C}^{n\times n}$, let $W({\bf A})$ be its field of values, and let $\EuScript P_k$ denote the polynomials of degree at most $k$. Define…

Gau, Wang and Wu's conjecture. For nonzero $a$, $W(F_{aI})$ should be an elliptical disk with minor half-axis

Gau et al.'s conjecture. The centre of $W(A)$ must be the origin; equivalently, $a=0$. This extends the known result for $n\leq 4$ to arbitrary dimension. The paper gives a partial…

Let $k$ be the relevant matrix size, let $\lambda\in\mathbb C$, let $\beta\in\mathbb C$ be nonzero, let $U$ be unitary, and let $B$ be a matrix whose field of values is contained i…

Let $\mathcal{P}_m$ be the space of complex polynomials of degree at most $m$, let $\mathcal M^n$ be the space of $n\times n$ complex matrices, and for $A\in\mathcal M^n$ let…

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…

Let $A\in\mathcal{M}_n(\mathbb{C})$ be an $n\times n$ complex matrix. Its quaternionic numerical range is…

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.