10 problems

Stampfli-type conjecture. If $(T_{1},T_{2})$ is hyponormal and $T_{1}^{m}T_{2}^{n}$ is normal for some $m\geq 1$ and $n\geq 1$, then $(T_{1},T_{2})$ is normal.

Perturbation stability conjecture. Under a sufficiently small perturbation of the weight sequence $\alpha$, the resulting weighted shift remains weakly $k$-hyponormal.

Curto–Muhly–Xia conjecture. The pair $\mathbf{T}$ is subnormal if and only if it is hyponormal.

Curto–Hwang–Lee conjecture. There exists a finite Blaschke–Potapov product $B\in\mathcal{E}(\Phi)$ such that

Let $W_{\mathcal A}$ be a cubically hyponormal matrix-valued weighted shift, and let $x\in\mathcal H$. Suppose that … for some $k\geq 1$. Backward propagation conjecture. Then … fo…

Derivative inequality conjecture. Under these hypotheses,

Conjecture on sums of self-adjoint and hyponormal symbols. The sum of a self-adjoint symbol and a hyponormal symbol should always correspond to a hyponormal Toeplitz operator; equi…

Let $N>4$ be a positive integer. The Cesàro matrix $C(N)$ has entries … and define the diagonal operator … Hyponormality conjecture for Cesàro matrices. The operator $C(N)$ is boun…

Let $M\in\mathcal{B}(\ell^2)$ be the generalized Cesàro matrix of order two with parameter $\alpha$. Hyponormality conjecture. The operator $M$ is also hyponormal for…

Matrix-valued Nakazi–Takahashi conjecture. If $\Phi\in L^\infty_{M_n}$ is such that $T_\Phi$ is hyponormal and its self-commutator $[T_\Phi^*,T_\Phi]$ has finite rank, then there e…