26 problems

Let $d4d8_p$ be the collection of symbols that are bounded Schur multipliers of the Schatten–von Neumann class $d4d4_p$, and let $d4d8_p^{\mathrm{cb}}$ be the collection of symbols…

Strong Schatten convergence conjecture. The convergence above is strong in every $\mathfrak{S}^p(\mathfrak{H}_k)$ with $p>1/s+1/2$.

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

Let $A_1,\ldots,A_n\in S_p$, where $S_p$ is the Schatten $p$-class. For $1<p<\infty$, let $q$ be the conjugate exponent of $p$, so that $\frac1p+\frac1q=1$. Audenaert–Kittaneh conj…

Let $S_p$ denote the Schatten $p$-class, and let $A,B,C\in S_p$. For $p>0$, write $\\|\cdot\\|_p$ for the Schatten $p$-quasi-norm. Sharp three-operator Clarkson–McCarthy conjecture…

Let $\mathscr{H}$ be a Hilbert space, let $\mathbb{B}_p(\mathscr{H})$ denote the Schatten $p$-class, and define $\omega_j=e^{ji\pi/n}$. Let…

Let $\mathscr{H}$ be a Hilbert space and let $\mathbb{B}_p(\mathscr{H})$ denote the Schatten $p$-class on $\mathscr{H}$. For $A,B\in\mathbb{B}_p(\mathscr{H})$, the noncommutative C…

Audenaert–Kittenah conjecture. For $1\le p\le 2$,

Mishra–Vemuri's Schatten-class conjecture. The operator $L_{\mathcal{F}_\sigma(\mu)}$ belongs to the Schatten class $\mathcal{S}^{p}$ if and only if

Audenaert–Kittaneh's conjecture. For $1<p\le 2$,

Ball–Carlen–Lieb conjecture. If $p\ge 2$, then for any $A,B\in S_p$,

Noncommutative Hanner inequality conjecture. The first inequality holds for all $p\geq 2$, and the reverse inequality holds for all $1\leq p\leq 2$.

Two-dimensional Schatten embedding conjecture. For any $p\geq 1$ and $A,B\in\mathcal{S}_p$, the real space

For $1\leq p\leq2\leq q\leq\infty$, define … Let $a_n$ denote the $n$th approximation number of the embedding. The piecewise approximation-number conjecture. There is a universal c…

Let $S_p^N$ and $S_q^N$ be Schatten classes of $N\times N$ matrices, and let $d_n$ denote the $n$th Kolmogorov number of the embedding. For $0<p\leq\infty$, $0<q\leq1$, $q\leq p$,…

The Gelfand–approximation conjecture. One has

GKS's conjecture. The approximation numbers satisfy

Conjecture. The following are equivalent:

Let $S^p$ be the Schatten–von Neumann class associated with a real Hilbert space, with $1<p<\infty$ and $p\notin\mathbb{N}$. Write $[p]$ for the integer part of $p$. Differentiabil…

Let $T=[T_1,dots,T_d]$ be a pure graded finite-rank $d$-contraction, and let ${\mathfrak S}_p$ denote the Schatten $p$-class of compact operators whose singular numbers lie in…

Let $(S_t)_{t\ge 0}$ be the semigroup of shifts on the semiaxis, let $(W_t)_{t\ge 0}$ be a cocycle, and define the cocyclic perturbation by $(V_t=W_tS_t)_{t\ge 0}$. Let…

Let $p\in\mathbb{C}[z_1,\ldots,z_n]$, let $Z(p)$ be its zero set, and let $\mathscr{Q}_p$ be the associated quotient module with operators $Q_f$. Consider the commutators of these…

Let $p\in\mathbb{C}[z_1,\ldots,z_n]$, let $\mathscr{Q}_p$ be the relevant quotient Hilbert module, and let $Q_f$ denote the operator induced by a polynomial or multiplier $f$ on…

Let $F_\alpha$ be the class of functions $f:\mathbb{R}\mapsto\mathbb{C}$ such that $f(a)-f(b)\in S^\alpha$ whenever $a$ and $b$ are self-adjoint and $a-b\in S^\alpha$, where…

Let $F_1$ be the class of functions $f:\mathbb{R}\mapsto\mathbb{C}$ such that $f(a)-f(b)\in S^1$ whenever $a$ and $b$ are self-adjoint and $a-b\in S^1$. Krein's conjecture. The con…