9 problems

Power hyponormality conjecture. If

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…

The conjecture. In Sector II, except on its boundary with Sector I, no $\alpha$ is interpolated by a Bernstein function; in Sector III, $W_\alpha$ is subnormal, and except on its b…

Let $T$ be the random weighted shift associated with independent weights $\{X_n\}_{n=1}^{\infty}$ on a Hilbert space $\mathcal{H}$, and let $\{e_k\}_{k=1}^{\infty}$ be its associat…

Let $A\in\mathcal{B}(\mathcal{H})$ be an expansive, convex analytic operator whose moment sequence has linear growth, meaning that there exist constants $c_1,c_2$ such that … for e…

Let $W_{(\alpha,\beta)}$ be a commuting $2$-variable weighted shift, and suppose that its toral and spherical Aluthge transforms, denoted by $\widetilde{W}_{(\alpha,\beta)}$ and…

Let $1\leq p<\infty$, and let $T$ be the bilateral weighted shift on $\ell^p(\mathbb{Z})$ defined by … for a bounded positive weight sequence $(w_j)_{j\in\mathbb{Z}}$. Weak hypercy…

Let $T$ be an irreducible weighted shift with weights $\{\alpha_n\}_{n=1}^{\infty}$. For each $n\in\mathbb{N}$ and $\epsilon>0$, suppose there is an index $c_{n,\epsilon}\geq n$ su…

Slice-measure inequalities conjecture. The subnormality of $\mathbf{T}$ is determined by a countable collection of inequalities