10 problems

Prăjitură's conjecture. The set $A_T$ is either empty or dense in $X$.

Let $U$ be a bounded linear operator on a Banach space. An operator is universal in the sense of Rota if its scalar multiples can approximate every continuous linear operator in th…

Let $\phi$ be an entire function of exponential type, and let $D$ denote differentiation on the space of entire functions. The convolution operator $\phi(D)$ is defined by the symb…

Let $X$ be a topological vector space and let $T$ be a continuous linear operator on $X$. For a finite set $F\subset X$, consider its orbit under $T$, namely…

Distributional-chaos conjecture. If $T$ is chaotic in the sense of Devaney, then $T$ is distributionally chaotic.

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…

Chaotic-operator conjecture. Every chaotic operator is frequently hypercyclic and admits an ergodic invariant measure with full support.

Let $X$ be a Banach space. For a bounded linear operator $R$ on $X$, define … Theorem 1 in the paper asserts that, for every infinite-dimensional separable real or complex Banach s…

Let $phi$ be a linear map on $\mathbb{R}^n$, and let an SAT mean a self-affine tiling with expansion map $phi$. There is an integer matrix $M$ acting on $\mathbb{R}^N$ for some…

Let $\phi$ be a linear expansion map for a self-affine tiling of $\mathbb{R}^n$. The paper establishes a necessary condition on $\phi$: assuming that $\phi$ is diagonalizable over…