10 problems

Infinite-complement conjecture. The set $T\setminus\varphi(T)$ should be infinite.

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…

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 operator, let $T^*$ be its adjoint, and let $d4dc_1$ and $d4dc_2$ be subspaces. Assume that $T$ is $d4dc_1$-transitive and $T^*$ is $d4dc_2$-transitive. Orthogonal-co…

Let $\ell_2({\mathbb Z}_+)$ be the complex Hilbert space of square-summable sequences indexed by $\mathbb Z_+$, and let $B^*$ be the backward shift on this space. Feldman's conject…

Feldman's conjecture. The adjoint operator $M^*$ is $1$-weakly hypercyclic on $H^2(G)$. The source presents this as a conjecture and gives no resolution.

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…

Hypercyclicity conjecture. Any separable ${\mathcal F}$-space with infinite-dimensional dual supports a hypercyclic operator.

Perfect spanning conjecture. If $T$ is a hypercyclic operator on $X$ whose eigenvectors associated to eigenvalues of modulus $1$ span a dense subspace of $X$, are the unimodular ei…