10 problems

Let $f_1,\ldots,f_l,g\in\mathbb{C}\langle\underline{x}\rangle$. For a complex Hilbert space $\mathcal{H}$, let $\mathcal{B}(\mathcal{H})$ denote the bounded operators on…

Let $\mathbb{K}$ be a field, let $f_1,\ldots,f_l,g\in\mathbb{K}\langle\underline{x}\rangle$ be noncommutative polynomials, and for each $n\in\mathbb{N}$ and…

Let $E_1$ and the relevant Hardy-space-valued symbols be as in Theorem 1, with $T_{\Phi}$ and $T_{\Theta}$ the associated Toeplitz operators. Define … Here a half-space is a closed…

Invariant-subspace certificate conjecture. The following are equivalent: (i) for every $n\in\mathbb{N}$ and every $\underline X\in\operatorname{M}_n(\mathbbm k)^d$, every joint inv…

Let $\alpha$ be a continuous $\|\cdot\|_1$-dominating normalized gauge norm, and let $H^{\alpha}$ and $M_{\alpha}(z^n)$ be the associated Hardy space and invariant subspace. Decomp…

Let $A^2_1$ be the Bergman space … A closed subspace of $A^2_1$ is singly generated if it is the minimal closed invariant subspace containing one function, where invariance is with…

Let $A^2_1$ be the weighted Bergman space of holomorphic functions on the unit disk with norm … For $H\in A^2_1$, write … the smallest closed subspace of $A^2_1$ containing $H$ and…

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 $G=P_n$ be the path graph on $n$ cells, and let $\mathcal D_{G,\ell}$ denote the collection of transformations whose invariant subspaces are being classified by the algorithm i…

Let $\mathcal{A}$ be a finite family of linear operators $\mathbb K^d\to\mathbb K^d$, where $\mathbb K$ is an arbitrary field and $d\geqslant 2$. Define $HI(\mathbb K^d)$ to be the…