21 problems

Let $X$ be a $K$-convex Banach space. The property $\mathrm{GL}_2$ is the Gordon–Lewis property with exponent $2$, and a Banach lattice has finite cotype in the usual Banach-space…

Let $X$ be a Banach space and let $1\le q<\infty$. The property $S\mathrm{GL}(1,q)$ is the hypothesis that $X$ has in the statement below. SGL embedding conjecture. If $X$ has…

Let $X$ be a Dedekind complete Banach lattice, and let $P\colon X\to X$ be a positive projection. Suppose there is an $b1\ge 0$ such that … and every positive operator $T$ satisfyi…

Countable-chain-condition conjecture. For every $p$-natural quasi-Banach space $E$, the space

Let $K$ be a compact space. A complemented subspace is the range of a bounded projection. Conjecture on $C[0,1]$ and $L_1[0,1]$. Every complemented subspace of $C[0,1]$ is isomorph…

Let $E$ be a Banach space. Say that $E$ has DPR-lust when it has the local unconditional structure introduced by Gordon, Lewis, and related work; a complemented subspace of a Banac…

Norm-attainment correspondence conjecture. The functional $x^*$ attains its norm if and only if the lattice homomorphism $\widehat{x^*}$ attains its norm.

Let $E$ be $L^p( Omega, mu)$ for $1leq pleq infty$, or more generally a Banach lattice, and let $(Tt){tin(0,infty)}$ be a mean ergodic $C0$-semigroup on $E$. A **quasi-interior fix…

Let $E$ be a Banach space and let $1{\leqslant} p<\infty$. Write $\operatorname{FBL}^p[E]$ for the free Banach lattice associated with $E$, and let $I_{w^*}[E]$ be the correspondin…

Fine-spectrum and compactness conjecture. The operator $\mathcal{C}_t$ is compact on $X$, and its point, continuous and residual spectra are respectively

Let $E$ be a Banach space, and let $\mathrm{FBL}[E]$ denote the free Banach lattice generated by $E$. A Banach space is weakly compactly generated (WCG) if it is the closed linear…

A complex Banach lattice is necessarily an appropriate complexification of a real Banach lattice. Complexification conjecture. The free complex Banach lattice should be the complex…

Rotation-invariance conjecture. The essential spectra

Peetre's isometry conjecture. The coincidence of $\left\langle A_0,A_1\right\rangle_\theta$ and $\left[A_0,A_1\right]_\theta$ for couples of lattices is an isometry when…

Let $X$ be a Banach lattice with Stonean compact $K$, let $Y$ be the subspace and $T$ the weighted composition operator from Proposition 1, and let $S$ be the corresponding weighte…

Let $X$ be a projective Banach lattice, and let an order continuous functional mean a continuous linear functional preserving limits of decreasing nets to zero in order. The atoms…

Let $\textswab{a}$ be an uncountable set, let $X$ be a projective Banach lattice, and suppose that $X$ contains a closed ideal isomorphic to the free Banach lattice…

Let $X$ be a projective Banach lattice, and let a topological order unit mean an element whose principal order ideal is dense in $X$. A Banach lattice is free when it is a free Ban…

A projective Banach lattice is a Banach lattice with the projective lifting property, and a topological order unit is an element whose principal order ideal is dense in the lattice…

Let $G$ be a locally compact abelian group, and let $X$ be a space of locally integrable functions on $G$ whose norm is shift-invariant. For $\mu\in M(G)$, where $M(G)$ is the alge…

Ercan–Önal conjecture. The conclusion that $X$ is homeomorphic to $Y$ and that $E$ and $F$ are isomorphic as Banach lattices should hold for any Banach lattice $F$, without assumin…