  We consider a Schr\"odinger operator with complex-valued potentials on the
line. The operator has essential spectrum on the half-line plus eigenvalues
(counted with algebraic multiplicity) in the complex plane without the positive
half-line. We determine series of trace formulas. Here we have the new term: a
singular measure, which is absent for real potentials. Moreover, we estimate of
sum of Im part of eigenvalues plus singular measure in terms of the norm of
potentials. The proof is based on classical results about the Hardy spaces.
