27 problems

Let $D^1$ denote the unit disc, let $k(z)=(1+alpha)(1-|z|^2)^\alpha$ with $\alpha>4$, and let $A^2(\mathbb{C},D^1,kd\mu)$ be the corresponding weighted Bergman space. The ARS chara…

Independence-relaxation conjecture. The column-wise independence condition may be relaxed to independence among the $E_{i,j}$'s, for $i,j\in\{1,2\}$.

Let $T$ be a linear differential operator acting on functions in ${\bold R}^k$. Suppose that $T$ possesses a single finite-dimensional invariant subspace with polynomial basis, pre…

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 $H$ be a complex separable Hilbert space with $\dim H=\infty$, and let $T:H\to H$ be a bounded linear operator. A subspace $V\subset H$ is non-trivial if it is proper, closed,…

Minimality conjecture. Under this assumption, $K_g$ is minimal if and only if $g$ is a $C_{\phi_a}$-eigenvector. This conjecture concerns the classification of minimal invariant su…

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…

Polynomial-bound conjecture. The bound $r\cdot (r+1)^{r+1}$ should be improvable to a polynomial (possibly linear) function of $r$, at least when $G$ is finite.

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…

An operator acts on an infinite-dimensional dual Banach space when it is a bounded linear operator on a Banach space that is the dual of another Banach space. Invariant subspace co…

Let $\cN \subset \cM \subset \cH_k \otimes \cE$ be two non-zero closed subspaces represented as $\cM = \Phi \cdot (\cH_s \otimes \cF)$ and $\cN = \Psi \cdot (\cH_s \otimes \cG)$ by…

Invariant subspace conjecture. Every bounded linear operator on a Banach space has a non-trivial closed invariant subspace; that is, there exists a closed subspace $M$ with…

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…

Mohebi–Radjabalipour conjecture. Under these conditions, $T$ has a nontrivial invariant subspace.

Let $G$ be a complete graph, and let $\mathcal D_{G,\ell}$ denote the collection of transformations whose invariant subspaces are under consideration. A subspace is a fixed point s…

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 $\mathcal{L}$ be a rank-one perturbation of a compact normal operator. An invariant subspace with a common spectral part is an invariant subspace $\mathcal{M}$ satisfying … for…

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…

Let $T$ be an operator on a Banach space. Operators with a compact square are operators for which $T^2$ is compact. Following the existence of invariant subspaces for compact opera…

Let $T$ be a bounded operator on a separable Hilbert space. A closed subspace $\mathcal{V}$ is non-trivial if $\mathcal{V}\ne\{0\}$ and $\mathcal{V}$ is not the whole Hilbert space…

Invariant subspace conjecture. Every bounded linear operator $T$ on a separable Hilbert space $H$ has a non-trivial closed invariant subspace.