  In this paper, we introduce and study the spectral property $(E_A)$. This
property means that the difference between the approximate point spectrum and
the upper semi-Fredholm spectrum coincides with the difference between the
approximate point spectrum and the upper semi-Weyl spectrum. Together with
local spectral theory, we explore the behavior of this property under certain
topological conditions and derive characterizations for the operators that
verify it. Furthermore, we establish sufficient conditions that guarantee that
a bounded linear operator verifies the property $(E_A)$.
