20 problems

Power hyponormality conjecture. If

Let $\mathbf{T}=(T_1,T_2)$ be a pair of commuting subnormal operators on a Hilbert space $\mathcal{H}$. Commuting subnormal-pair conjecture. Then $\mathbf{T}$ is subnormal if and o…

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

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…

Let $\mu$ be a finitely atomic Berger measure with support in $\mathbb{R}_+$. Curto's conjecture. The following statements are equivalent. This conjecture concerns the relationship…

Let $W_\mu$ be a recursively generated (i.e., $\operatorname{card}\operatorname{supp}\mu<\infty$) subnormal weighted shift. Curto's conjecture. The following statements are equival…

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…

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

Let $W_\mu$ be a subnormal weighted shift with finitely atomic Berger measure $\mu$. The measure $\mu$ has a square root if there exists a nonnegative measure $\nu$ such that … The…

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 $f(z)=\sum_{n=0}^{\infty}a_nz^n$ be a function in the random weighted-shift setting of Theorem, with $\sigma^2<\infty$ as in that theorem. Conjecture. The condition … is unnece…

For $0 \le t \le \frac{1}{2}$, the generalized mean transform of a linear operator $T$ is defined by … where $\widetilde{T}(t):=|T|^tU|T|^{1-t}$ is the generalized Aluthge transfor…

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…

Polynomial-exponent bound conjecture. There exist positive constants $a,b$ such that

Adjoint subspace-transitivity conjecture. $T$ is $\mathcal{M}$-transitive if and only if $T^*$ is $\mathcal{M}^{\perp}$-transitive. This would relate subspace-transitivity of an op…

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 a continuous linear operator on a separable Banach space $X$. Assume that every $x\in X$ has a backward orbit converging to zero in norm, and that every $x$ in some dens…

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