  We use the principle of almost orthogonality to give a new and simple proof
that a sparse Lerner operator is bounded on a matrix- or operator-weighted
space $L_W^{2}(\mu)$, where $\mu$ is a doubling measure on $\R^d$ if and only
if the weight $W$ satisfies the Muckenhoupt $A_2(\mu)$-condition, restricted to
the sparse collection in question. Our method extends to the
infinite-dimensional setting, thus allowing for applications to the
multi-parameter setting. For the class of Muckenhoupt $A_2$-weights, we obtain
bounds in terms of mixed $A_{2}(\mu)$-$A_{\infty}(\mu)$-conditions, which is
independent of dimension and agrees with the best known bound in the
finite-dimensional vectorial setting. As an application, we prove a
matrix-weighted bound for the maximal Bergman projection, where we obtain a new
sharper bound in terms of the B\'ekoll\'e-Bonami characteristic. Furthermore,
we consider commutators of sparse Lerner operators on operator-valued weighted
$L^{2}$-spaces and some applications to multi-parameters
