  It follows, from a generalised version of Paley-Wiener theorem, that the
Laplace transform is an isometry between certain spaces of weighted $L^2$
functions defined on $(0, \infty)$ and (Hilbert) spaces of analytic functions
on the right complex half-plane (for example Hardy, Bergman or Dirichlet
spaces). We can use this fact to investigate properties of multipliers and
multiplication operators on the latter type of spaces. In this paper we present
a full characterisation of multipliers in terms of a generalised concept of a
Carleson measure. Under certain conditions, these spaces of analytic functions
are not only Hilbert spaces but also Banach algebras, and are therefore
contained within their spaces of multipliers. We provide some necessary as well
as sufficient conditions for this to happen and look at its consequences.
