  We study principal component analysis (PCA) for mean zero i.i.d. Gaussian
observations $X_1,\dots, X_n$ in a separable Hilbert space $\mathbb{H}$ with
unknown covariance operator $\Sigma.$ The complexity of the problem is
characterized by its effective rank ${\bf r}(\Sigma):= \frac{{\rm
tr}(\Sigma)}{\|\Sigma\|},$ where ${\rm tr}(\Sigma)$ denotes the trace of
$\Sigma$ and $\|\Sigma\|$ denotes its operator norm. We develop a method of
bias reduction in the problem of estimation of linear functionals of
eigenvectors of $\Sigma.$ Under the assumption that ${\bf r}(\Sigma)=o(n),$ we
establish the asymptotic normality and asymptotic properties of the risk of the
resulting estimators and prove matching minimax lower bounds, showing their
semi-parametric optimality.
