15 problems

Functional-calculus conjecture. It should be possible to extend this functional calculus beyond the w-continuous module maps to a homomorphism from all of $CB_{d4d5}(B(d4d4))$ into…

Generalized monomial Carleson-frame conjecture. The sequence

Let $X_1,\ldots,X_r$ be spectral operators in the sense of Dunford on separable Hilbert spaces $\mathcal{H}_1,\ldots,\mathcal{H}_r$. Assume there are positive self-adjoint operator…

Let $d\in\mathbb{N}$, let $p$ and $\gamma$ be parameters for which the weighted Sobolev spaces below are defined, let $X$ be the underlying Banach space, and let…

Crouzeix's conjecture. $K_{\mathrm{bfd}}=2$.

Let $G$ be a Lie group of polynomial volume growth, generated by vector fields $X_1,\ldots,X_m$, and let $\slashed{D}_p$ be the $L^p$ realization of the Hodge–Dirac operator on … F…

Let $G$ be a connected Lie group of polynomial growth equipped with a family of left-invariant Hörmander vector fields, let $\Delta_p$ be the associated Laplacian on…

Let $G$ be a connected Lie group of polynomial growth equipped with a family $(X_1,\ldots,X_m)$ of left-invariant Hörmander vector fields, let $\slashed{D}_p$ be the associated unb…

Let $T_t$ be a Markovian semigroup satisfying the technical hypotheses of Corollary and let $A$ be its generator. Let $w$ be a $Q^A_2$ weight. An angle $\theta\in(0,\pi)$ is a univ…

Let $A\in\mathbb C^{d,d}$, let $W(A)=\{v^*Av;\ v\in\mathbb C^d,\ \|v\|^2=v^*v=1\}$ be its numerical range, and let $\|\cdot\|$ denote the operator norm. For a polynomial…

Let $X$ be a $C^k$-manifold and $E$ a complex $C^k$ vector bundle of rank $r$ on $X$. Let $(y^1,\ldots,y^n)$ be a global coordinate system on $\Bbb R^n$, and let … be an assignment…

Let $H$ be a Hilbert space, let $L_1(H)$ denote the trace-class operators on $H$, and let $L_{1,cinfty}(H)$ denote the weak-$L_1$ space of compact operators $T$ whose singular valu…

Let $H$ and $A$ belong to a $C^{*}$-algebra, with $H$ a positive contraction satisfying $0\leq H\leq1$ and $\left\Vert A\right\Vert\leq1$. For the square-root function…

Let $\mathcal G \rightrightarrows \mathrm M$ be a groupoid with compact units $\mathrm M$ and polynomial growth. The holomorphic functional calculus conjecture asserts that the exp…

Spectrum-domain conjecture. Every ${AC}(\sigma)$ operator is in fact an ${AC}(\sigma(T))$ operator.