Asymptotics for isotropic Hilbert-valued spherical random fields

In this paper, we introduce the concept of isotropic Hilbert-valued spherical random field, thus extending the notion of isotropic spherical random field to an infinite-dimensional setting. We then establish a spectral representation theorem and a functional Schoenberg's theorem. Following some key results established for the real-valued case, we prove consistency and quantitative central limit theorem for the sample power spectrum operators in the high-frequency regime.