Context. Because of its proximity to an active K-type star, the hot Jupiter WASP-80b has been identified as a possible excellent target for detecting and measuring He I absorption in the upper atmosphere. Aims. Our aim was to look for, and eventually measure and model, metastable He I atmospheric absorption. Methods. We observed four primary transits of WASP-80b in the optical and near-infrared using the HARPS-N and GIANO-B high-resolution spectrographs attached to the Telescopio Nazionale Galileo telescope, focusing the analysis on the He I triplet. We further employed a three-dimensional hydrodynamic aeronomy model to understand the observational results. Results. We did not find any signature of planetary absorption at the position of the He I triplet with an upper limit of 0.7% (i.e. 1.11 planetary radii; 95% confidence level). We re-estimated the high-energy stellar emission, which we combined with a stellar photospheric model, to generate the input for the hydrodynamic modelling. We determined that, assuming a solar He to H abundance ratio, He I absorption should have been detected. Considering a stellar wind 25 times weaker than solar, we could reproduce the non-detection only by assuming a He to H abundance ratio about 16 times smaller than solar. Instead, considering a stellar wind ten times stronger than solar, we could reproduce the non-detection only with a He to H abundance ratio about ten times smaller than solar. We attempted to understand this result by collecting all past He I measurements and looking for correlations with high-energy stellar emission and planetary gravity, but without success. Conclusions. WASP-80b is not the only planet with an estimated sub-solar He to H abundance ratio, which suggests the presence of efficient physical mechanisms (e.g. phase separation, magnetic fields) capable of significantly modifying the He to H content in the upper atmosphere of hot Jupiters. The planetary macroscopic properties and the shape of the stellar spectral energy distribution are not sufficient for predicting the presence or absence of detectable metastable He in a planetary atmosphere, since the He abundance also appears to play a major role.

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Abstract. We characterize injective maps preserving -order and maps strongly preserving < -order on the set of diagonalizable matrices.Mathematics subject classification (2010): 15A86, 15A04, 15A09.