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Let $X$ and $Y$ be compact hyper-Kahler manifolds deformation equivalence to the Hilbert scheme of length $n$ subschemes of a $K3$ surface. A cohomology class in their product $X\times Y$ is an analytic correspondence, if it belongs to the subring generated by Chern classes of coherent analytic sheaves. Let $f$ be a Hodge isometry of their second rational cohomologies with respect to the Beauville-Bogomolov-Fujiki pairings. We prove that $f$ is induced by an analytic correspondence. We furthermore lift $f$ to an analytic correspondence $F$ between their total rational cohomologies, which is a Hodge isometry with respect to the Mukai pairings, and which preserves the gradings up to sign. When $X$ and $Y$ are projective the correspondences $f$ and $F$ are algebraic.
