  Analytic Langlands correspondence was proposed by Etingof, Frenkel and
Kazhdan. On one side of this correspondence there are certain operators on
$L^2(\operatorname{Bun}_G)$, called Hecke operators, where
$\operatorname{Bun}_G$ is the variety of stable $G$-bundles on $X$ and
$L^2(\operatorname{Bun}_G)$ is a Hilbert space of square-integrable
half-densities. The compactness conjecture says that Hecke operators are
bounded and, moreover, compact. In arXiv:2106.05243 Etingof, Frenkel and
Kazhdan prove this and other conjectures in the case of
$G=\operatorname{PGL}_2$ and $X=\mathbb{P}^1$ with parabolic structures. We
investigate the case of $G=\operatorname{PGL}_2$ and genus one curve over
complex numbers with parabolic structures. We obtain an explicit formula for
Hecke operators and prove the compactness conjecture in this case.
