  Let $X,X_1,\dots, X_n$ be i.i.d. Gaussian random variables in a separable
Hilbert space ${\mathbb H}$ with zero mean and covariance operator
$\Sigma={\mathbb E}(X\otimes X),$ and let $\hat \Sigma:=n^{-1}\sum_{j=1}^n
(X_j\otimes X_j)$ be the sample (empirical) covariance operator based on
$(X_1,\dots, X_n).$ Denote by $P_r$ the spectral projector of $\Sigma$
corresponding to its $r$-th eigenvalue $\mu_r$ and by $\hat P_r$ the empirical
counterpart of $P_r.$ The main goal of the paper is to obtain tight bounds on
$$ \sup_{x\in {\mathbb R}} \left|{\mathbb P}\left\{\frac{\|\hat
P_r-P_r\|_2^2-{\mathbb E}\|\hat P_r-P_r\|_2^2}{{\rm Var}^{1/2}(\|\hat
P_r-P_r\|_2^2)}\leq x\right\}-\Phi(x)\right|, $$ where $\|\cdot\|_2$ denotes
the Hilbert--Schmidt norm and $\Phi$ is the standard normal distribution
function. Such accuracy of normal approximation of the distribution of squared
Hilbert--Schmidt error is characterized in terms of so called effective rank of
$\Sigma$ defined as ${\bf r}(\Sigma)=\frac{{\rm
tr}(\Sigma)}{\|\Sigma\|_{\infty}},$ where ${\rm tr}(\Sigma)$ is the trace of
$\Sigma$ and $\|\Sigma\|_{\infty}$ is its operator norm, as well as another
parameter characterizing the size of ${\rm Var}(\|\hat P_r-P_r\|_2^2).$ Other
results include non-asymptotic bounds and asymptotic representations for the
mean squared Hilbert--Schmidt norm error ${\mathbb E}\|\hat P_r-P_r\|_2^2$ and
the variance ${\rm Var}(\|\hat P_r-P_r\|_2^2),$ and concentration inequalities
for $\|\hat P_r-P_r\|_2^2$ around its expectation.
