  Let $\hat\Sigma=\frac{1}{n}\sum_{i=1}^n X_i\otimes X_i$ denote the sample
covariance operator of centered i.i.d.~observations $X_1,\dots,X_n$ in a real
separable Hilbert space, and let $\Sigma=\mathbb{E}(X_1\otimes X_1)$. The focus
of this paper is to understand how well the bootstrap can approximate the
distribution of the operator norm error $\sqrt
n\|\hat\Sigma-\Sigma\|_{\text{op}}$, in settings where the eigenvalues of
$\Sigma$ decay as $\lambda_j(\Sigma)\asymp j^{-2\beta}$ for some fixed
parameter $\beta>1/2$. Our main result shows that the bootstrap can approximate
the distribution of $\sqrt n\|\hat\Sigma-\Sigma\|_{\text{op}}$ at a rate of
order $n^{-\frac{\beta-1/2}{2\beta+4+\epsilon}}$ with respect to the Kolmogorov
metric, for any fixed $\epsilon>0$. In particular, this shows that the
bootstrap can achieve near $n^{-1/2}$ rates in the regime of large $\beta$ --
which substantially improves on previous near $n^{-1/6}$ rates in the same
regime. In addition to obtaining faster rates, our analysis leverages a
fundamentally different perspective based on coordinate-free techniques.
Moreover, our result holds in greater generality, and we propose a model that
is compatible with both elliptical and Mar\v{c}enko-Pastur models in
high-dimensional Euclidean spaces, which may be of independent interest.
