  This paper develops a new Hilbert space method to characterize a family of
reproducing kernel Hilbert spaces of real harmonic functions in a bounded
Lipschitz domain $\Omega \subset \mathbb R^d, d\geq 2$ involving some families
of positive self-adjoint operators and making use of characterizations of their
trace data and of a special inner product on $H^1(\Omega).$ We also establish
boundary representation results for this family in terms of the $L^2-$ Bergman
kernel. In particular, a boundary integral representation for the very weak
solution of the Dirichlet problem for Laplace's equation with $L^2- $ boundary
data is provided. Reproducing kernels and orthonormal bases for the harmonic
spaces are also found.
