  The aim of the present paper is three folds. For a reproducing kernel Hilbert
space $\mathcal{A}$ (R.K.H.S) and a $\sigma-$finite measure space
$(M_{1},d\mu_{1})$ for which the corresponding $L^{2}-$space is a separable
Hilbert space, we first build an isometry of Bargmann type as an integral
transform from $L^{2}(M_{1},d\mu_{1})$ into $\mathcal{A}$. Secondly, in the
case where there exists a $\sigma-$finite measure space $(M_{2},d\mu_{2})$ such
that the Hilbert space $L^{2}(M_{2},d\mu_{2})$ is separable and
$\mathcal{A}\subset L^{2}(M_{2},d\mu_{2})$ the inverse isometry is also given
in an explicit form as an integral transform. As consequence, we recover some
classical isometries of Bargmann type. Thirdly, for the classical Dirichlet
space as R.K.H.S, we elaborate a new isometry of Bargmann type. Furthermore,
for this Dirichlet space, we give a new characterization, as harmonic space of
a single second order elliptic partial differential operator for which, we
present some spectral properties. Finally, we extend the same results to a
class of generalized Bergman-Dirichlet space.
