  We develop direct and inverse spectral analysis for finite and semi-infinite
non-self-adjoint Jacobi matrices with a rank one imaginary part. It is shown
that given a set of $n$ not necessarily distinct non-real numbers in the open
upper (lower) half-plane uniquely determines a n x n Jacobi matrix with a rank
one imaginary part having those numbers as its eigenvalues counting
multiplicity. An algorithm for reconstruction for such finite Jacobi matrices
is presented. A new model complementing the well known Livsic triangular model
for bounded linear operators with rank one imaginary part is obtained. It turns
out that the model operator is a non-self-adjoint Jacobi matrix and it follows
from the fact that any bounded, prime, non-self-adjoint linear operator with
rank one imaginary part acting on some finite-dimensional (resp., separable
infinite-dimensional Hilbert space) is unitary equivalent to a finite (resp.,
semi-infinite) non-self-adjoint Jacobi matrix. This result strengthens the
classical Stone theorem established for self-adjoint operators with simple
spectrum. We establish the non-self-adjoint analogs of the Hochstadt and
Gesztesy--Simon uniqueness theorems for finite Jacobi matrices with non-real
eigenvalues as well as an extension and refinement of these theorems for finite
non-self-adjoint tri-diagonal matrices to the case of mixed eigenvalues, real
and non-real. A unique Jacobi matrix, unitarily equivalent to the operator of
indefinite integration in the Hilbert space L_2[0,l] is found as well as
spectral properties of its perturbations and connections with well known
Bernoulli numbers. We also give the analytic characterization of the Weyl
functions of dissipative Jacobi matrices with a rank one imaginary part.
