  We consider the estimation of quadratic functionals in a Gaussian sequence
model where the eigenvalues are supposed to be unknown and accessible through
noisy observations only. Imposing smoothness assumptions both on the signal and
the sequence of eigenvalues, we develop a minimax theory for this problem. We
propose a truncated series estimator and show that it attains the optimal rate
of convergence if the truncation parameter is chosen appropriately.
Consequences for testing problems in inverse problems are equally discussed: in
particular, the minimax rates of testing for signal detection and
goodness-of-fit testing are derived.
