  Recently it has been shown that any regular simple symmetric operator with
deficiency indices (1,1) is unitarily equivalent to the operator of
multiplication in a reproducing kernel Hilbert space of functions on the real
line with the Kramer sampling property. This work has been motivated, in part,
by potential applications to signal processing and mathematical physics. In
this paper we exploit well-known results about de Branges-Rovnyak spaces and
characteristic functions of symmetric operators to prove that any such a
symmetric operator is in fact unitarily equivalent to multiplication by the
independent variable in a de Branges space of entire functions. This leads to
simple new results on the spectra of such symmetric operators, on when
multiplication by z is densely defined in de Branges-Rovnyak spaces in the
upper half plane, and to sufficient conditions for there to be an isometry from
a given subspace of $L^2 (R, dv) onto a de Branges space of entire functions
which acts as multiplication by a measurable function.
