483 problems

Let $A$ be an $n\times n$ matrix, let $0<|q|\leq 1$, and let $\Omega_q$ denote the scaled $q$-numerical range introduced in the source. For a polynomial $p$ with complex coefficien…

Let the $d$-shift module be the Hilbert module over the polynomial ring on the complex unit ball, and let a submodule be homogeneous when it is generated by homogeneous elements. A…

Let $H^2$ be the Hardy space on the unit disk $\mathbb D$, let $f,g\in H^2$, and let $T_f$ and $T_{\overline g}$ denote the corresponding Toeplitz operators on $H^2$. For…

Self-adjointness conjecture. $T$ is self-adjoint.

For an operator $A\in\mathcal{B}(\mathcal{H})$ with polar decomposition $A=U|A|$, define its Aluthge transform by … Write $\Delta^n(A)$ for the $n$-fold iterate, and let the strong…

Kadison–Singer conjecture. There is a partition $X_1,\ldots,X_r$ of $[m]$ such that

For a set $S\subseteq\mathbb{N}$, let $P_S$ be the projection onto the coordinates indexed by $S$, and let $E(T)$ denote the diagonal operator of an operator…

Let $\mathbf{T}$ and $\widetilde{\mathbf{T}}$ be commuting operator tuples in a Cowen–Douglas class $\mathbf{\mathcal{B}}_{n}^{m}(\Omega)$, with associated Hermitian holomorphic ei…

Halmos's conjecture. For every contraction $T\in B(H)$,

Let $D$ denote the Dirichlet space, let $L^2_a$ denote the Bergman space, and let $M_\phi$ be multiplication by a finite Blaschke product $\phi$ of order $n$ on $L^2_a$. A reducing…

Gau–Wang–Wu conjecture. If the numerical range $W(A)$ is a circular disc centered at the origin, then $A$ is rotationally invariant.

For $0<p<\infty$ and $\alpha>-1$, let $A^p_\alpha$ be the weighted Bergman space of holomorphic functions on the unit disk $\mathbb{D}$ with … where…

Let $\mathbb D$ be the unit disc and $H^2(\mathbb D)$ its Hardy space. For an analytic self-map $\varphi$ of $\mathbb D$, define the composition operator $C_\varphi$ by … Let…

Let $\mathcal{H}$ be a Hilbert space, let $A$ satisfy $0<mI\leq A\leq MI$, and let $X,Y$ be partial isometries on $\mathcal{H}$ whose final spaces are orthogonal. For every 2-posit…

Geometric Arveson–Douglas conjecture. The quotient module $Q$ is $p$-essentially normal for every $p>\dim M$.

Let $X=(X_{ij})_{1\leq i,j\leq n}$ be the random matrix in the setting of the section, with independent centered Gaussian entries as specified there, and let $\|X\|$ denote its ope…

Koplienko's conjecture. There exists a real Borel measure $\nu_p=\nu_{p,H_0,V}$ whose total variation satisfies

Let $T$ be an $N\times N$ normal matrix, and let $k\geq 1$ be a fixed positive integer. Write $\Lambda_k(T)$ for the rank-$k$ numerical range of $T$, and let $\Gamma$ range over al…

Let $A_1$ and $A_2$ be $n\times n$ Hermitian matrices, and define their characteristic polynomial by … Suppose that $P_A$ has a repeated factor in the polynomial ring…

Let $X=\mathbb{C}^n$ be equipped with an arbitrary norm. An operator $T\in B(X)$ satisfies the Bhatia–Šemrl property if, for every $A\in B(X)$ with $T\perp_B A$, there exists a uni…

Let $H^2(\mathbb{D}^2)$ be the Hardy space on the bidisk, and let $M$ be a submodule. Define … where $R_z$ and $R_w$ are multiplication by the coordinate functions on $M$. The subm…

For $1<p<\infty$, let $P$ denote the Riesz projection from $L^p$ onto the classical Hardy space $H^p$, and let $\mathcal{B}(L^p)$ be the bounded operators on $L^p$. Gohberg–Krupnik…

Paszkiewicz's conjecture. The sequence $(S_n)$ converges strongly.

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

Non-closability conjecture. The sum $\mathcal{L}_{1,\mu}+\mathcal{L}_2$ is not closable.