39 problems

Let $\mathcal{PH}_\alpha^2(\mathbb{C}^n)$ be the pluriharmonic Fock space, let $\mu$ be a positive Borel measure, and let $T_\mu$ be the associated Toeplitz operator. For $r>0$, wr…

König's conjecture. For each Banach space $F$ with non-trivial type, there exists an $s$ with $2<s<\infty$ such that

Limit-order conjecture.

Injective total accessibility conjecture. Then $\mathcal A^{\operatorname{inj}}$ is totally accessible.

Realification non-surjectivity conjecture. There exists a complex operator ideal which cannot be obtained from a real operator ideal by this construction.

Weak Schatten estimate conjecture. One has

Let $H$ be a closed subspace of $C(K)$, where $K$ is a compact Hausdorff space, and let $B_{H^{*}}$ denote the closed unit ball of the dual space $H^{*}$. A Lipschitz $p$-…

The paper considers more general Hörmander classes $S_0(M)$ and super operators acting on Hilbert-Schmidt operators. Super-operator extension conjecture. Extending these Hörmander…

Let … of $^n$, and let $1<pbackslashle 2$. Let $A^p$ denote the Bergman space, let $T$ be the Toeplitz operator induced by … be its Berezin transform, and let $K(,)$ be the Bergma…

Let $(\mathbf{E},\|\cdot\|_{\mathbf{E}})$ be a Banach space with a $1$-symmetric basis, and let $\mathsf{S}_{\mathbf{E}}$ be the corresponding unitary ideal. Assume that…

Endpoint Sobolev absolute summability conjecture. The inclusion

Absolute summability conjecture for exact interpolation spaces. The embedding

Endpoint absolute summability conjecture. The inclusion

Kalton's conjecture. Every Banach space that does not contain $\ell_1^n$ uniformly complemented is a Banach $\mathcal K$-space. Equivalently, this condition should be sufficient fo…

Let $X$ and $Y$ be real Banach spaces. The pair $(X,Y)$ has the Hilbert space factorization property (HFP) when every bounded operator $\phi:X\to Y$ satisfies…

Let $1<p<2$, set $r=2p/(2-p)$, and let $i_{p2}\colon\ell_p\to\ell_2$ be the formal identity map. Let $\pi_{r,2}(\cdot)$ denote the $(r,2)$-absolutely summing norm. For a linear map…

Mityagin–Henkin conjecture. The factor $n^2$ can be replaced by $n$, namely,

GKS's conjecture. The approximation numbers satisfy

Let $L(X,Y)$ denote the bounded linear operators between Banach spaces $X$ and $Y$. An operator ideal $U$ is proper if its associated space ideal … is the class of finite-dimension…

Let $G$ be a compact abelian group, let $Z(G)$ be an invariant function space, and let $T:H_1\to H_2$ be an operator between Hilbert spaces. Say that $T$ factorizes invariantly thr…

Let $W^{1}_{1}$ denote the Sobolev space on the relevant torus, and let $T$ be an operator between Hilbert spaces that factorizes through $W^{1}_{1}$. Invariant little Grothendieck…

Let $\partial\Omega$ be a $C^{\infty}$ simply connected closed surface, let $\mathcal{K}_{\partial\Omega}$ be its Neumann–Poincaré operator, and let $\zeta(s)$ denote the Riemann z…

Let $1<p<\finite$ and let $\xi$ be an ordinal. The operator ideal $\mathfrak{Sl}_p^\xi$ is said to have the $\mathfrak{Sl}_p$ factorization property when every operator in…

Completely continuous polynomial ideal conjecture. The class $\mathcal{CC}\circ\mathcal{P}$ is not a polynomial two-sided ideal.

Schatten-class symbol conjecture. For every $p>2$ there exists a multiplicative Hankel form $H_\varphi$ in $S_p(\mathscr{H}^2)$ without a bounded symbol.