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Let $X,X_1,\dots, X_n$ be i.i.d. Gaussian random variables with zero mean andcovariance operator $\Sigma={\mathbb E}(X\otimes X)$ taking values in aseparable Hilbert space ${\mathbb H}.$ Let $$ {\bf r}(\Sigma):=\frac{{\rmtr}(\Sigma)}{\|\Sigma\|_{\infty}} $$ be the effective rank of $\Sigma,$ ${\rmtr}(\Sigma)$ being the trace of $\Sigma$ and $\|\Sigma\|_{\infty}$ being itsoperator norm. Let $$\hat \Sigma_n:=n^{-1}\sum_{j=1}^n (X_j\otimes X_j)$$ bethe sample (empirical) covariance operator based on $(X_1,\dots, X_n).$ Thepaper deals with a problem of estimation of spectral projectors of thecovariance operator $\Sigma$ by their empirical counterparts, the spectralprojectors of $\hat \Sigma_n$ (empirical spectral projectors). The focus is onthe problems where both the sample size $n$ and the effective rank ${\bfr}(\Sigma)$ are large. This framework includes and generalizes well knownhigh-dimensional spiked covariance models. Given a spectral projector $P_r$corresponding to an eigenvalue $\mu_r$ of covariance operator $\Sigma$ and itsempirical counterpart $\hat P_r,$ we derive sharp concentration bounds forbilinear forms of empirical spectral projector $\hat P_r$ in terms of samplesize $n$ and effective dimension ${\bf r}(\Sigma).$ Building upon theseconcentration bounds, we prove the asymptotic normality of bilinear forms ofrandom operators $\hat P_r -{\mathbb E}\hat P_r$ under the assumptions that$n\to \infty$ and ${\bf r}(\Sigma)=o(n).$ In a special case of eigenvalues ofmultiplicity one, these results are rephrased as concentration bounds andasymptotic normality for linear forms of empirical eigenvectors. Other resultsinclude bounds on the bias ${\mathbb E}\hat P_r-P_r$ and a method of biasreduction as well as a discussion of possible applications to statisticalinference in high-dimensional principal component analysis.

Asymptotics and Concentration Bounds for Bilinear Forms of Spectral Projectors of Sample Covariance