  When a resonance lies near the threshold of a heavier channel, an interesting
feature can occur. The paradigmatic example employed here is the scalar
isoscalar $f_0(980)$ resonance that couples to the lighter $\pi\pi$ and heavier
$K\bar{K}$ channels. It is shown that the decay width is given by the sum or
subtraction of the partial decay widths depending on whether the pole lies in
the Riemann sheet that is contiguous with the physical one above or below the
$K\bar{K}$ threshold, respectively. Next, we show that the usually disregarded
renormalization of bare parameters in Flatt\'e or energy-dependent Breit-Wigner
parameterizations is essential to extract physical information. The
compositeness of the $f_0(980)$ by using a Flatt\'e parameterization matched to
reproduce the pole properties obtained from Roy equations and other analytic
constraints is evaluated.
