  In this paper, we consider the family of planar piecewise linear differential
systems with two zones separated by a straight line without sliding regions,
that is, differential systems whose flow transversally crosses the switching
line except for at most one point. In the research literature, many papers deal
with the problem of determining the maximum number of limit cycles that these
differential systems can have. This problem has been usually approached via
large case-by-case analyses which distinguish the many different possibilities
for the spectra of the matrices of the differential systems. Here, by using a
novel integral characterization of Poincar\'e half-maps, we prove, without
unnecessary distinctions of matrix spectra, that the optimal uniform upper
bound for the number of limit cycles of these differential systems is one. In
addition, it is proven that this limit cycle, if it exists, is hyperbolic and
its stability is determined by a simple condition in terms of the parameters of
the system. As a byproduct of our analysis, a condition for the existence of
the limit cycle is also derived.
