  Let $E$ be a separable Banach space and let $X, X_1,\dots, X_n, \dots$ be
i.i.d. Gaussian random variables taking values in $E$ with mean zero and
unknown covariance operator $\Sigma: E^{\ast}\mapsto E.$ The complexity of
estimation of $\Sigma$ based on observations $X_1,\dots, X_n$ is naturally
characterized by the so called effective rank of $\Sigma:$ ${\bf r}(\Sigma):=
\frac{{\mathbb E}_{\Sigma}\|X\|^2}{\|\Sigma\|},$ where $\|\Sigma\|$ is the
operator norm of $\Sigma.$ Given a smooth real valued functional $f$ defined on
the space $L(E^{\ast},E)$ of symmetric linear operators from $E^{\ast}$ into
$E$ (equipped with the operator norm), our goal is to study the problem of
estimation of $f(\Sigma)$ based on $X_1,\dots, X_n.$ The estimators of
$f(\Sigma)$ based on jackknife type bias reduction are considered and the
dependence of their Orlicz norm error rates on effective rank ${\bf
r}(\Sigma),$ the sample size $n$ and the degree of H\"older smoothness $s$ of
functional $f$ are studied. In particular, it is shown that, if ${\bf
r}(\Sigma)\lesssim n^{\alpha}$ for some $\alpha\in (0,1)$ and $s\geq
\frac{1}{1-\alpha},$ then the classical $\sqrt{n}$-rate is attainable and, if
$s> \frac{1}{1-\alpha},$ then asymptotic normality and asymptotic efficiency of
the resulting estimators hold. Previously, the results of this type (for
different estimators) were obtained only in the case of finite dimensional
Euclidean space $E={\mathbb R}^d$ and for covariance operators $\Sigma$ whose
spectrum is bounded away from zero (in which case, ${\bf r}(\Sigma)\asymp d$).
