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.
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