  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.
