26 problems

Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$, and let $B,B_2,\ldots,B_m\in M(n,\mathbb{K})$. Let $B^*$ denote the conjugate transpose and let $[A,B]=AB-BA$. Assume … for…

Let $T_X:M(n,\mathbb{C})\to M(n,\mathbb{C})$ be defined by $T_X(Y)=[X^*,[X,Y]]$, and let $\lambda_1(T_X)\geq\cdots\geq\lambda_N(T_X)$ be its eigenvalues. Ge–Li–Lu–Zhou conjecture.…

Let $T_X:M(n,\mathbb{C})\to M(n,\mathbb{C})$ be defined by $T_X(Y)=[X^*,[X,Y]]$, and let $\lambda_1(T_X)\geq\cdots\geq\lambda_N(T_X)$ be its eigenvalues. Lu–Wenzel eigenvalue conje…

Let $\mathbb{K}=\mathbb{R}$ and let $B,B_2,\ldots,B_m\in M(n,\mathbb{K})$. Let $B^*$ denote the conjugate transpose, let $\bot$ denote orthogonality for the Frobenius inner produc…

Let $\mathbb{K}=\mathbb{R}$ and let $B_1,\ldots,B_m\in M(n,\mathbb{K})$, where $M(n,\mathbb{K})$ is the space of $n\times n$ matrices over $\mathbb{K}$. Let $[A,B]=AB-BA$ and let…

Commutator Baer–Suzuki conjecture. If $[x,g]$ is either $1$ or $p$-singular for every element $g\in G$, then $x\in O_p(G)$.

Let $\operatorname{CL}_{\lVert\!\lVert\!\lVert\cdot\rVert\!\rVert\!\rVert}([0,\infty))$ denote the class of commutator-Lipschitz functions for the chosen unitarily invariant norm,…

Let $\widehat{\mathcal M}_+$ be the space of Fourier transforms of finite positive Borel measures on $\mathbb R$, and define … Let $f$ be a non-negative function that is operator m…

Let $A$, $B$, and $X$ be operators in $B({\mathcal H})$. Assume that $A,B\geq 0$ and that $X$ is compact with finite unitarily invariant norm…

Let $A$, $B$, and $X$ be operators in $B({\mathcal H})$. Assume that $A,B\geq 0$ are compact with finite unitarily invariant norm…

Hecke deformation conjecture. Corollary 28 and Corollary 29 both seem to hold in $\mathcal{H}$ when every $t_\ell$ is replaced by $t_\ell^{\mathcal{H}}$.

Let $P$ and $Q$ be the momentum and position operators, and let $f$ and $g$ be bounded, real, measurable functions. Define … For a positive parameter $\alpha$, let $K_{\alpha}$ den…

Let $L$ be a non-nilpotent regular Lie algebra with a nondegenerate symmetric invariant bilinear form. An element $x\in L$ is called a commutant element if $x$ belongs to the commu…

Kitaev's determinant conjecture. Under these assumptions, the Fredholm determinant satisfies

Let $T$ be a non-degenerate Calderón–Zygmund operator with Dini-continuous kernel, let $m\in\mathbb N$ and $p>1$, and let $\lambda$, $\mu$, and $\eta$ be arbitrary weights. Here…

Let $b_1,b_2$ satisfy the same assumptions as in Theorems and. Let $H$ denote the Hilbert transform, let $A,B,C$ be Young functions, let $\bar A,\bar B,\bar C$ be their complementa…

Let $M(n,\mathbb{R})$ be the space of real $n\times n$ matrices. Let $B,B_2,\ldots,B_m\in M(n,\mathbb{R})$, with $B_\alpha^*$ denoting transpose and with the Frobenius inner produc…

Sharpness conjecture. If $(m,n)\ne(3,3)$, then there is a unital associative ring $A$ and elements $a_i,b_j\in A$ such that

Let $A$ be an unbounded self-adjoint operator on a Hilbert space. Unbounded commutator conjecture. There is always a bounded self-adjoint operator $B$ such that $AB-BA$ is unbounde…

Let $G$ be a finite non-abelian simple group, and let $g\in G$. A commutator is an element of the form $xyx^{-1}y^{-1}$ for some $x,y\in G$. Ore's conjecture. Every element…

Etingof–Kim–Ma conjecture.

General trace commutator conjecture. There exists a constant $c$, independent of $d$, such that

Let $\mathcal{X}$ be a Banach space such that … where $1\leq p\leq\infty$ or $p=0$ (we say that such a space admits a Pełczyński decomposition). Assume that…

Schatten-norm commutator conjecture. In the restricted case $p=q$,

Makar-Limanov and Jie-Tai Yu's conjecture. For every $m>n$ not divisible by $n$, the formal power series