  This paper studies the learning of linear operators between
infinite-dimensional Hilbert spaces. The training data comprises pairs of
random input vectors in a Hilbert space and their noisy images under an unknown
self-adjoint linear operator. Assuming that the operator is diagonalizable in a
known basis, this work solves the equivalent inverse problem of estimating the
operator's eigenvalues given the data. Adopting a Bayesian approach, the
theoretical analysis establishes posterior contraction rates in the infinite
data limit with Gaussian priors that are not directly linked to the forward map
of the inverse problem. The main results also include learning-theoretic
generalization error guarantees for a wide range of distribution shifts. These
convergence rates quantify the effects of data smoothness and true eigenvalue
decay or growth, for compact or unbounded operators, respectively, on sample
complexity. Numerical evidence supports the theory in diagonal and non-diagonal
settings.
