5 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.

Let $A$ be a normal operator on a complex Hilbert space of dimension $n\ge 2$. Let $\rho(A)$ denote the radius of the smallest closed disk containing the spectrum of $A$, let $A'$…

Normalized-frame conjecture. The system above is not a frame for $\mathcal{H}$.

Davidson–Szarek conjecture. There is a universal constant $C$ such that

Davies–Simon conjecture. The estimate holds with a function $F$ satisfying