348 problems

Let $G$ be a Gromov hyperbolic group, and consider affine actions of $G$ on a Hilbert space whose linear parts are uniformly bounded representations. An affine action is proper whe…

Bader–Furman–Gelander–Monod's conjecture. Higher-rank algebraic groups should possess Banach property (T) with respect to the class of all super-reflexive Banach spaces.

Let $\ell_1^n$ denote $n$-dimensional real space with its usual $\ell_1$ norm, and let $e(X)$ be the maximum cardinality of an equilateral set in a metric space $X$. The canonical…

Banach's isometric embedding conjecture. $L_p$ isometrically embeds into $L_r$.

An infinite-dimensional Banach space $X$ is minimal if every infinite-dimensional subspace of $X$ has a further subspace isomorphic to $X$. Rosenthal's conjecture. Every infinite-d…

Optimality conjecture. The bound

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 $\mathcal{X}$ be a Banach space. Let $A\colon\Omega\to\mathcal{L}(\mathcal{X})$ be a generator of a semigroup $e^{At}$ such that $\lVert A\rVert$ is bounded, and let…

Let $\mathbb{X}$ be a Banach space with a unique unconditional basis. For $E\in\{c_{0},\ell_{1},\ell_{2}\}$, consider the iterated copy $E(\mathbb{X})$ of $\mathbb{X}$, namely the…

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

Let $X$ be a Banach space and let $F\in\mathcal{B}(X)$. The perturbation property in question is invariance of the descent spectrum under commuting perturbations by $F$, when $F^n$…

Let $\mathfrak Z_n$ be the Rochberg space, and let $\mathfrak L(\mathfrak Z_n)$ denote its bounded operators. A Fredholm operator is an operator with finite-dimensional kernel and…

Let $({\mathcal M},\rho)$ be a pseudometric space, let $X$ be a Banach space, and let $m$ be a fixed positive integer. Set … For a set-valued mapping…

Let $T$ be a Kreiss bounded operator on a Hilbert space, meaning that there is a constant $C>0$ such that … for all $n\geq 1$. Shields' conjecture. Every Kreiss bounded Hilbert-spa…

Let $(T_t)_{t\geqslant 0}$ be a weak continuous semigroup of selfadjoint contractive operators on $\mathrm{L}^\infty(\Omega)$, where $\Omega$ is a measure space. A Banach space $X$…

Complemented subspace conjecture. Every complemented subspace of $L_1[0,1]$ is isomorphic either to $\ell_1$ or to $L_1[0,1]$.

Wood's conjecture. If $C_0(L,\mathbb{K})$ is almost transitive, then $L$ consists of a single point.

Krivine's conjecture. The bound obtained from Krivine's original scheme is exactly the value of the Grothendieck constant $K_G$.

Metric cotype growth conjecture. Every Banach space with Rademacher cotype $q<\infty$ satisfies the metric-cotype inequality with an even integer $m\lesssim_X n^{1/q}$, without ass…

Let $X$ be the Banach space from Proposition 3.16, which is an $M$-ideal in $X^{**}$ but admits no $M$-approximate identity in $X^{**}$. Separable intermediate-space conjecture. Th…

Let $X$ be a finite-dimensional Banach space, let $q$ be in the range assumed in the surrounding discussion, let $X^*$ have cotype constant $C_q(X^*)$, and let $w_2(X)$ be the seco…

Let $w_2(\ell_{\infty}^n)$ denote the second Whitney constant associated with the $n$-dimensional cube $Q^n$, and let $E_2(f;Q^n)$ and $\omega_2(f:Q^n)$ denote the corresponding qu…

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.

Let $(X_0,X_1)$ be a Banach couple, and call a Banach space an interpolation space for the couple when it is an intermediate space associated with $(X_0,X_1)$ in the sense of inter…