160 problems

Foci conjecture. If $\operatorname{cl}(W(T))$ is an elliptical disk, then its foci are contained in $\{0,1,-1\}$.

For each $n\geq 1$, define $S_n$ to be the least constant such that, for every $n$-dimensional complex Banach space $X$ and every invertible operator $T\in\mathcal{L}(X)$,…

Let $\mathcal{H}$ be a complex Hilbert space and let $U,V\in\mathcal{B}(\mathcal{H})$ be invertible. If $(U-\mathbf{1})(V-\mathbf{1})$ and $(V-\mathbf{1})(U-\mathbf{1})$ belong to…

For every integer $n\ge 1$, every matrix $A\in\mathbb{C}^{n\times n}$, and every polynomial $p\in\mathbb{C}[z]$, one has $\|p(A)\|_2\le 2\sup_{z\in W(A)}|p(z)|$, where…

For every finitely generated polynomial submodule $M\subset H^2(\mathbb D^2)$, Yang's numerical invariants satisfy $\Sigma_0(M)\geq \Sigma_1(M)\geq \Sigma_2(M)\geq\cdots$.

For Hermitian matrices $A_{1},A_{2}\in M_{n}(\mathbb{C})$, define $p(z_{0},z_{1},z_{2})=\det\!\left(z_{0}I+z_{1}A_{1}+z_{2}A_{2}\right)$. Kippenhahn's conjecture asserts that if th…

Let $\alpha(\omega)$ denote the norm of $W(-\nu_{0}+i\omega)$, where $W$ is the operator-valued function defined in the preceding construction. Asymptotic almost periodicity conjec…

Essential positivity conjecture. The operator $T_f$ is essentially positive if and only if

Let $\mathcal{N}(\mathbb{C}^n)$ denote the set of nilpotent operators on $\mathbb{C}^n$, and let $\delta_n$ be the distance from the set of non-zero projections in…

Let $A$ be the generator of a $C_0$-group on $\ell_p$, with $p>1$, whose eigenvalues, counted with multiplicity, are $\{\lambda_n\}_{n=1}^{\infty}$ and whose corresponding normaliz…

Spectral radius extension conjecture. The spectral radius formulae of Theorem $$ are valid for all $\varphi\in\mathcal{S}_m$.

Let $G$ be an abstract completely continuous operator, let $\Lambda:=\left\{\lambda_j\right\}$ be the set of its eigenvalues, and fix $\delta>0$. For each $\lambda$, define the circle…

Power hyponormality conjecture. If

Asymptotic constant conjecture. There exists a constant $1\leq c\leq\sqrt e$ such that

Let $\{\alpha_n\}_{n\in\mathbb{N}}$ be unimodular complex numbers and let $\{\beta_n\}_{n\in\mathbb{N}}$ be positive numbers converging to $0$, with the ratios…

Let $\{z_n\}_{n\in\mathbb{N}}$ be distinct points in the open unit disk converging to $1$. Let $\mathcal{H}$ be a separable infinite-dimensional complex Hilbert space with orthonor…

Let $I\subseteq\mathbb{C}[\mathbf{z}]$ be an ideal, let $\mathcal{Z}$ be its zero variety, and let $\mathcal{S}$ be the submodule obtained by taking the closure of $I$ in…

Let $\mathcal{M}$ be a finite-rank, quasi-free, $p$-reductive Hilbert module over $A(\Omega)$, and let $\mathcal{S}$ be a submodule such that $\mathcal{S}^{\bot}$ is spanned by gen…

Let $\Omega$ be a bounded, strongly pseudoconvex domain in $\mathbb{C}^m$, and consider submodules of $B^2(\Omega)\otimes\mathbb{C}^k$. Finite-multiplicity p-reductivity conjecture…

Let $\Omega$ be a bounded, strongly pseudoconvex domain in $\mathbb{C}^m$ with smooth boundary, let $B^2(\Omega)$ be its Bergman space, and let $\mathcal{S}\subseteq B^2(\Omega)$ b…

Curto–Muhly–Xia conjecture. The pair $\mathbf{T}$ is subnormal if and only if it is hyponormal.

Direct-summand conjecture. The modules $\mathcal{R}_1$ and $\mathcal{R}_2$ are quasi-free of ranks $k_1$ and $k_2$, respectively, and

Embedding conjecture. The natural embedding $E\subset L_1([0,1])$ is not strictly singular if and only if $G$ embeds continuously into $E$.

Zeno product formula conjecture. If assumption in Theorem II.6a is dropped, then for every $f\in{\mathfrak h}$, $T>0$, and arbitrary admissible function $\phi$,

Let $M$ be a complex symmetric matrix, and suppose that $M$ is unitarily equivalent to a truncated Toeplitz operator (TTO). Let $B$ be an inner function, let $K_B$ be the correspon…