5 problems

Perälä–Virtanen conjecture. The operator $T_f$ is essentially positive if and only if

Let $\Gamma$ be a piecewise $C^3$ Jordan curve with finitely many corners, whose angles lie strictly between $0$ and $\pi$. Let $K \colon H^{1/2}(\Gamma) \to H^{1/2}(\Gamma)$ be th…

Let $M$ be a complete, non-compact Riemannian manifold with Ricci curvature bounded below and uniformly sub-exponential volume growth. For $p\geq 1$, the $L^p$-essential spectrum i…

Sub-exponential volume growth conjecture. The $L^p$ essential spectrum of $M$ is $[0,+\infty)$ for any $p\in[1,+\infty]$.

Massive essential spectrum conjecture. There exists a measurable function $p:\mathbb{T}\to(1,\infty)$ such that $S$ is bounded on $L^{p(\cdot)}(\mathbb{T})$ and…